FieldVector3D.java

/*
 * Licensed to the Apache Software Foundation (ASF) under one or more
 * contributor license agreements.  See the NOTICE file distributed with
 * this work for additional information regarding copyright ownership.
 * The ASF licenses this file to You under the Apache License, Version 2.0
 * (the "License"); you may not use this file except in compliance with
 * the License.  You may obtain a copy of the License at
 *
 *      https://www.apache.org/licenses/LICENSE-2.0
 *
 * Unless required by applicable law or agreed to in writing, software
 * distributed under the License is distributed on an "AS IS" BASIS,
 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 * See the License for the specific language governing permissions and
 * limitations under the License.
 */

/*
 * This is not the original file distributed by the Apache Software Foundation
 * It has been modified by the Hipparchus project
 */

package org.hipparchus.geometry.euclidean.threed;

import java.io.Serializable;
import java.text.NumberFormat;

import org.hipparchus.CalculusFieldElement;
import org.hipparchus.Field;
import org.hipparchus.analysis.polynomials.SmoothStepFactory;
import org.hipparchus.exception.LocalizedCoreFormats;
import org.hipparchus.exception.MathIllegalArgumentException;
import org.hipparchus.exception.MathRuntimeException;
import org.hipparchus.geometry.LocalizedGeometryFormats;
import org.hipparchus.util.FastMath;
import org.hipparchus.util.FieldBlendable;
import org.hipparchus.util.FieldSinCos;
import org.hipparchus.util.MathArrays;

/**
 * This class is a re-implementation of {@link Vector3D} using {@link CalculusFieldElement}.
 * <p>Instance of this class are guaranteed to be immutable.</p>
 * @param <T> the type of the field elements
 */
public class FieldVector3D<T extends CalculusFieldElement<T>> implements FieldBlendable<FieldVector3D<T>, T>, Serializable {

    /** Serializable version identifier. */
    private static final long serialVersionUID = 20130224L;

    /** Abscissa. */
    private final T x;

    /** Ordinate. */
    private final T y;

    /** Height. */
    private final T z;

    /** Simple constructor.
     * Build a vector from its coordinates
     * @param x abscissa
     * @param y ordinate
     * @param z height
     * @see #getX()
     * @see #getY()
     * @see #getZ()
     */
    public FieldVector3D(final T x, final T y, final T z) {
        this.x = x;
        this.y = y;
        this.z = z;
    }

    /** Simple constructor.
     * Build a vector from its coordinates
     * @param v coordinates array
     * @exception MathIllegalArgumentException if array does not have 3 elements
     * @see #toArray()
     */
    public FieldVector3D(final T[] v) throws MathIllegalArgumentException {
        if (v.length != 3) {
            throw new MathIllegalArgumentException(LocalizedCoreFormats.DIMENSIONS_MISMATCH,
                                                   v.length, 3);
        }
        this.x = v[0];
        this.y = v[1];
        this.z = v[2];
    }

    /** Simple constructor.
     * Build a vector from its azimuthal coordinates
     * @param alpha azimuth (&alpha;) around Z
     *              (0 is +X, &pi;/2 is +Y, &pi; is -X and 3&pi;/2 is -Y)
     * @param delta elevation (&delta;) above (XY) plane, from -&pi;/2 to +&pi;/2
     * @see #getAlpha()
     * @see #getDelta()
     */
    public FieldVector3D(final T alpha, final T delta) {
        FieldSinCos<T> sinCosAlpha = FastMath.sinCos(alpha);
        FieldSinCos<T> sinCosDelta = FastMath.sinCos(delta);
        this.x = sinCosAlpha.cos().multiply(sinCosDelta.cos());
        this.y = sinCosAlpha.sin().multiply(sinCosDelta.cos());
        this.z = sinCosDelta.sin();
    }

    /** Multiplicative constructor.
     * Build a vector from another one and a scale factor.
     * The vector built will be a * u
     * @param a scale factor
     * @param u base (unscaled) vector
     */
    public FieldVector3D(final T a, final FieldVector3D<T>u) {
        this.x = a.multiply(u.x);
        this.y = a.multiply(u.y);
        this.z = a.multiply(u.z);
    }

    /** Multiplicative constructor.
     * Build a vector from another one and a scale factor.
     * The vector built will be a * u
     * @param a scale factor
     * @param u base (unscaled) vector
     */
    public FieldVector3D(final T a, final Vector3D u) {
        this.x = a.multiply(u.getX());
        this.y = a.multiply(u.getY());
        this.z = a.multiply(u.getZ());
    }

    /** Multiplicative constructor.
     * Build a vector from another one and a scale factor.
     * The vector built will be a * u
     * @param a scale factor
     * @param u base (unscaled) vector
     */
    public FieldVector3D(final double a, final FieldVector3D<T> u) {
        this.x = u.x.multiply(a);
        this.y = u.y.multiply(a);
        this.z = u.z.multiply(a);
    }

    /** Linear constructor.
     * Build a vector from two other ones and corresponding scale factors.
     * The vector built will be a1 * u1 + a2 * u2
     * @param a1 first scale factor
     * @param u1 first base (unscaled) vector
     * @param a2 second scale factor
     * @param u2 second base (unscaled) vector
     */
    public FieldVector3D(final T a1, final FieldVector3D<T> u1,
                         final T a2, final FieldVector3D<T> u2) {
        final T prototype = a1;
        this.x = prototype.linearCombination(a1, u1.getX(), a2, u2.getX());
        this.y = prototype.linearCombination(a1, u1.getY(), a2, u2.getY());
        this.z = prototype.linearCombination(a1, u1.getZ(), a2, u2.getZ());
    }

    /** Linear constructor.
     * Build a vector from two other ones and corresponding scale factors.
     * The vector built will be a1 * u1 + a2 * u2
     * @param a1 first scale factor
     * @param u1 first base (unscaled) vector
     * @param a2 second scale factor
     * @param u2 second base (unscaled) vector
     */
    public FieldVector3D(final T a1, final Vector3D u1,
                         final T a2, final Vector3D u2) {
        final T prototype = a1;
        this.x = prototype.linearCombination(u1.getX(), a1, u2.getX(), a2);
        this.y = prototype.linearCombination(u1.getY(), a1, u2.getY(), a2);
        this.z = prototype.linearCombination(u1.getZ(), a1, u2.getZ(), a2);
    }

    /** Linear constructor.
     * Build a vector from two other ones and corresponding scale factors.
     * The vector built will be a1 * u1 + a2 * u2
     * @param a1 first scale factor
     * @param u1 first base (unscaled) vector
     * @param a2 second scale factor
     * @param u2 second base (unscaled) vector
     */
    public FieldVector3D(final double a1, final FieldVector3D<T> u1,
                         final double a2, final FieldVector3D<T> u2) {
        final T prototype = u1.getX();
        this.x = prototype.linearCombination(a1, u1.getX(), a2, u2.getX());
        this.y = prototype.linearCombination(a1, u1.getY(), a2, u2.getY());
        this.z = prototype.linearCombination(a1, u1.getZ(), a2, u2.getZ());
    }

    /** Linear constructor.
     * Build a vector from three other ones and corresponding scale factors.
     * The vector built will be a1 * u1 + a2 * u2 + a3 * u3
     * @param a1 first scale factor
     * @param u1 first base (unscaled) vector
     * @param a2 second scale factor
     * @param u2 second base (unscaled) vector
     * @param a3 third scale factor
     * @param u3 third base (unscaled) vector
     */
    public FieldVector3D(final T a1, final FieldVector3D<T> u1,
                         final T a2, final FieldVector3D<T> u2,
                         final T a3, final FieldVector3D<T> u3) {
        final T prototype = a1;
        this.x = prototype.linearCombination(a1, u1.getX(), a2, u2.getX(), a3, u3.getX());
        this.y = prototype.linearCombination(a1, u1.getY(), a2, u2.getY(), a3, u3.getY());
        this.z = prototype.linearCombination(a1, u1.getZ(), a2, u2.getZ(), a3, u3.getZ());
    }

    /** Linear constructor.
     * Build a vector from three other ones and corresponding scale factors.
     * The vector built will be a1 * u1 + a2 * u2 + a3 * u3
     * @param a1 first scale factor
     * @param u1 first base (unscaled) vector
     * @param a2 second scale factor
     * @param u2 second base (unscaled) vector
     * @param a3 third scale factor
     * @param u3 third base (unscaled) vector
     */
    public FieldVector3D(final T a1, final Vector3D u1,
                         final T a2, final Vector3D u2,
                         final T a3, final Vector3D u3) {
        final T prototype = a1;
        this.x = prototype.linearCombination(u1.getX(), a1, u2.getX(), a2, u3.getX(), a3);
        this.y = prototype.linearCombination(u1.getY(), a1, u2.getY(), a2, u3.getY(), a3);
        this.z = prototype.linearCombination(u1.getZ(), a1, u2.getZ(), a2, u3.getZ(), a3);
    }

    /** Linear constructor.
     * Build a vector from three other ones and corresponding scale factors.
     * The vector built will be a1 * u1 + a2 * u2 + a3 * u3
     * @param a1 first scale factor
     * @param u1 first base (unscaled) vector
     * @param a2 second scale factor
     * @param u2 second base (unscaled) vector
     * @param a3 third scale factor
     * @param u3 third base (unscaled) vector
     */
    public FieldVector3D(final double a1, final FieldVector3D<T> u1,
                         final double a2, final FieldVector3D<T> u2,
                         final double a3, final FieldVector3D<T> u3) {
        final T prototype = u1.getX();
        this.x = prototype.linearCombination(a1, u1.getX(), a2, u2.getX(), a3, u3.getX());
        this.y = prototype.linearCombination(a1, u1.getY(), a2, u2.getY(), a3, u3.getY());
        this.z = prototype.linearCombination(a1, u1.getZ(), a2, u2.getZ(), a3, u3.getZ());
    }

    /** Linear constructor.
     * Build a vector from four other ones and corresponding scale factors.
     * The vector built will be a1 * u1 + a2 * u2 + a3 * u3 + a4 * u4
     * @param a1 first scale factor
     * @param u1 first base (unscaled) vector
     * @param a2 second scale factor
     * @param u2 second base (unscaled) vector
     * @param a3 third scale factor
     * @param u3 third base (unscaled) vector
     * @param a4 fourth scale factor
     * @param u4 fourth base (unscaled) vector
     */
    public FieldVector3D(final T a1, final FieldVector3D<T> u1,
                         final T a2, final FieldVector3D<T> u2,
                         final T a3, final FieldVector3D<T> u3,
                         final T a4, final FieldVector3D<T> u4) {
        final T prototype = a1;
        this.x = prototype.linearCombination(a1, u1.getX(), a2, u2.getX(), a3, u3.getX(), a4, u4.getX());
        this.y = prototype.linearCombination(a1, u1.getY(), a2, u2.getY(), a3, u3.getY(), a4, u4.getY());
        this.z = prototype.linearCombination(a1, u1.getZ(), a2, u2.getZ(), a3, u3.getZ(), a4, u4.getZ());
    }

    /** Linear constructor.
     * Build a vector from four other ones and corresponding scale factors.
     * The vector built will be a1 * u1 + a2 * u2 + a3 * u3 + a4 * u4
     * @param a1 first scale factor
     * @param u1 first base (unscaled) vector
     * @param a2 second scale factor
     * @param u2 second base (unscaled) vector
     * @param a3 third scale factor
     * @param u3 third base (unscaled) vector
     * @param a4 fourth scale factor
     * @param u4 fourth base (unscaled) vector
     */
    public FieldVector3D(final T a1, final Vector3D u1,
                         final T a2, final Vector3D u2,
                         final T a3, final Vector3D u3,
                         final T a4, final Vector3D u4) {
        final T prototype = a1;
        this.x = prototype.linearCombination(u1.getX(), a1, u2.getX(), a2, u3.getX(), a3, u4.getX(), a4);
        this.y = prototype.linearCombination(u1.getY(), a1, u2.getY(), a2, u3.getY(), a3, u4.getY(), a4);
        this.z = prototype.linearCombination(u1.getZ(), a1, u2.getZ(), a2, u3.getZ(), a3, u4.getZ(), a4);
    }

    /** Linear constructor.
     * Build a vector from four other ones and corresponding scale factors.
     * The vector built will be a1 * u1 + a2 * u2 + a3 * u3 + a4 * u4
     * @param a1 first scale factor
     * @param u1 first base (unscaled) vector
     * @param a2 second scale factor
     * @param u2 second base (unscaled) vector
     * @param a3 third scale factor
     * @param u3 third base (unscaled) vector
     * @param a4 fourth scale factor
     * @param u4 fourth base (unscaled) vector
     */
    public FieldVector3D(final double a1, final FieldVector3D<T> u1,
                         final double a2, final FieldVector3D<T> u2,
                         final double a3, final FieldVector3D<T> u3,
                         final double a4, final FieldVector3D<T> u4) {
        final T prototype = u1.getX();
        this.x = prototype.linearCombination(a1, u1.getX(), a2, u2.getX(), a3, u3.getX(), a4, u4.getX());
        this.y = prototype.linearCombination(a1, u1.getY(), a2, u2.getY(), a3, u3.getY(), a4, u4.getY());
        this.z = prototype.linearCombination(a1, u1.getZ(), a2, u2.getZ(), a3, u3.getZ(), a4, u4.getZ());
    }

    /** Build a {@link FieldVector3D} from a {@link Vector3D}.
     * @param field field for the components
     * @param v vector to convert
     */
    public FieldVector3D(final Field<T> field, final Vector3D v) {
        this.x = field.getZero().add(v.getX());
        this.y = field.getZero().add(v.getY());
        this.z = field.getZero().add(v.getZ());
    }

    /** Get null vector (coordinates: 0, 0, 0).
     * @param field field for the components
     * @return a new vector
     * @param <T> the type of the field elements
     */
    public static <T extends CalculusFieldElement<T>> FieldVector3D<T> getZero(final Field<T> field) {
        return new FieldVector3D<>(field, Vector3D.ZERO);
    }

    /** Get first canonical vector (coordinates: 1, 0, 0).
     * @param field field for the components
     * @return a new vector
     * @param <T> the type of the field elements
     */
    public static <T extends CalculusFieldElement<T>> FieldVector3D<T> getPlusI(final Field<T> field) {
        return new FieldVector3D<>(field, Vector3D.PLUS_I);
    }

    /** Get opposite of the first canonical vector (coordinates: -1, 0, 0).
     * @param field field for the components
     * @return a new vector
     * @param <T> the type of the field elements
     */
    public static <T extends CalculusFieldElement<T>> FieldVector3D<T> getMinusI(final Field<T> field) {
        return new FieldVector3D<>(field, Vector3D.MINUS_I);
    }

    /** Get second canonical vector (coordinates: 0, 1, 0).
     * @param field field for the components
     * @return a new vector
     * @param <T> the type of the field elements
     */
    public static <T extends CalculusFieldElement<T>> FieldVector3D<T> getPlusJ(final Field<T> field) {
        return new FieldVector3D<>(field, Vector3D.PLUS_J);
    }

    /** Get opposite of the second canonical vector (coordinates: 0, -1, 0).
     * @param field field for the components
     * @return a new vector
     * @param <T> the type of the field elements
     */
    public static <T extends CalculusFieldElement<T>> FieldVector3D<T> getMinusJ(final Field<T> field) {
        return new FieldVector3D<>(field, Vector3D.MINUS_J);
    }

    /** Get third canonical vector (coordinates: 0, 0, 1).
     * @param field field for the components
     * @return a new vector
     * @param <T> the type of the field elements
     */
    public static <T extends CalculusFieldElement<T>> FieldVector3D<T> getPlusK(final Field<T> field) {
        return new FieldVector3D<>(field, Vector3D.PLUS_K);
    }

    /** Get opposite of the third canonical vector (coordinates: 0, 0, -1).
     * @param field field for the components
     * @return a new vector
     * @param <T> the type of the field elements
     */
    public static <T extends CalculusFieldElement<T>> FieldVector3D<T> getMinusK(final Field<T> field) {
        return new FieldVector3D<>(field, Vector3D.MINUS_K);
    }

    /** Get a vector with all coordinates set to NaN.
     * @param field field for the components
     * @return a new vector
     * @param <T> the type of the field elements
     */
    public static <T extends CalculusFieldElement<T>> FieldVector3D<T> getNaN(final Field<T> field) {
        return new FieldVector3D<>(field, Vector3D.NaN);
    }

    /** Get a vector with all coordinates set to positive infinity.
     * @param field field for the components
     * @return a new vector
     * @param <T> the type of the field elements
     */
    public static <T extends CalculusFieldElement<T>> FieldVector3D<T> getPositiveInfinity(final Field<T> field) {
        return new FieldVector3D<>(field, Vector3D.POSITIVE_INFINITY);
    }

    /** Get a vector with all coordinates set to negative infinity.
     * @param field field for the components
     * @return a new vector
     * @param <T> the type of the field elements
     */
    public static <T extends CalculusFieldElement<T>> FieldVector3D<T> getNegativeInfinity(final Field<T> field) {
        return new FieldVector3D<>(field, Vector3D.NEGATIVE_INFINITY);
    }

    /** Get the abscissa of the vector.
     * @return abscissa of the vector
     * @see #FieldVector3D(CalculusFieldElement, CalculusFieldElement, CalculusFieldElement)
     */
    public T getX() {
        return x;
    }

    /** Get the ordinate of the vector.
     * @return ordinate of the vector
     * @see #FieldVector3D(CalculusFieldElement, CalculusFieldElement, CalculusFieldElement)
     */
    public T getY() {
        return y;
    }

    /** Get the height of the vector.
     * @return height of the vector
     * @see #FieldVector3D(CalculusFieldElement, CalculusFieldElement, CalculusFieldElement)
     */
    public T getZ() {
        return z;
    }

    /** Get the vector coordinates as a dimension 3 array.
     * @return vector coordinates
     * @see #FieldVector3D(CalculusFieldElement[])
     */
    public T[] toArray() {
        final T[] array = MathArrays.buildArray(x.getField(), 3);
        array[0] = x;
        array[1] = y;
        array[2] = z;
        return array;
    }

    /** Convert to a constant vector without extra field parts.
     * @return a constant vector
     */
    public Vector3D toVector3D() {
        return new Vector3D(x.getReal(), y.getReal(), z.getReal());
    }

    /** Get the L<sub>1</sub> norm for the vector.
     * @return L<sub>1</sub> norm for the vector
     */
    public T getNorm1() {
        return x.abs().add(y.abs()).add(z.abs());
    }

    /** Get the L<sub>2</sub> norm for the vector.
     * @return Euclidean norm for the vector
     */
    public T getNorm() {
        // there are no cancellation problems here, so we use the straightforward formula
        return x.square().add(y.square()).add(z.square()).sqrt();
    }

    /** Get the square of the norm for the vector.
     * @return square of the Euclidean norm for the vector
     */
    public T getNormSq() {
        // there are no cancellation problems here, so we use the straightforward formula
        return x.square().add(y.square()).add(z.square());
    }

    /** Get the L<sub>&infin;</sub> norm for the vector.
     * @return L<sub>&infin;</sub> norm for the vector
     */
    public T getNormInf() {
        return FastMath.max(FastMath.abs(x), FastMath.max(FastMath.abs(y), FastMath.abs(z)));
    }

    /** Get the azimuth of the vector.
     * @return azimuth (&alpha;) of the vector, between -&pi; and +&pi;
     * @see #FieldVector3D(CalculusFieldElement, CalculusFieldElement)
     */
    public T getAlpha() {
        return y.atan2(x);
    }

    /** Get the elevation of the vector.
     * @return elevation (&delta;) of the vector, between -&pi;/2 and +&pi;/2
     * @see #FieldVector3D(CalculusFieldElement, CalculusFieldElement)
     */
    public T getDelta() {
        return z.divide(getNorm()).asin();
    }

    /** Add a vector to the instance.
     * @param v vector to add
     * @return a new vector
     */
    public FieldVector3D<T> add(final FieldVector3D<T> v) {
        return new FieldVector3D<>(x.add(v.x), y.add(v.y), z.add(v.z));
    }

    /** Add a vector to the instance.
     * @param v vector to add
     * @return a new vector
     */
    public FieldVector3D<T> add(final Vector3D v) {
        return new FieldVector3D<>(x.add(v.getX()), y.add(v.getY()), z.add(v.getZ()));
    }

    /** Add a scaled vector to the instance.
     * @param factor scale factor to apply to v before adding it
     * @param v vector to add
     * @return a new vector
     */
    public FieldVector3D<T> add(final T factor, final FieldVector3D<T> v) {
        return new FieldVector3D<>(x.getField().getOne(), this, factor, v);
    }

    /** Add a scaled vector to the instance.
     * @param factor scale factor to apply to v before adding it
     * @param v vector to add
     * @return a new vector
     */
    public FieldVector3D<T> add(final T factor, final Vector3D v) {
        return new FieldVector3D<>(x.add(factor.multiply(v.getX())),
                y.add(factor.multiply(v.getY())),
                z.add(factor.multiply(v.getZ())));
    }

    /** Add a scaled vector to the instance.
     * @param factor scale factor to apply to v before adding it
     * @param v vector to add
     * @return a new vector
     */
    public FieldVector3D<T> add(final double factor, final FieldVector3D<T> v) {
        return new FieldVector3D<>(1.0, this, factor, v);
    }

    /** Add a scaled vector to the instance.
     * @param factor scale factor to apply to v before adding it
     * @param v vector to add
     * @return a new vector
     */
    public FieldVector3D<T> add(final double factor, final Vector3D v) {
        return new FieldVector3D<>(x.add(factor * v.getX()),
                y.add(factor * v.getY()),
                z.add(factor * v.getZ()));
    }

    /** Subtract a vector from the instance.
     * @param v vector to subtract
     * @return a new vector
     */
    public FieldVector3D<T> subtract(final FieldVector3D<T> v) {
        return new FieldVector3D<>(x.subtract(v.x), y.subtract(v.y), z.subtract(v.z));
    }

    /** Subtract a vector from the instance.
     * @param v vector to subtract
     * @return a new vector
     */
    public FieldVector3D<T> subtract(final Vector3D v) {
        return new FieldVector3D<>(x.subtract(v.getX()), y.subtract(v.getY()), z.subtract(v.getZ()));
    }

    /** Subtract a scaled vector from the instance.
     * @param factor scale factor to apply to v before subtracting it
     * @param v vector to subtract
     * @return a new vector
     */
    public FieldVector3D<T> subtract(final T factor, final FieldVector3D<T> v) {
        return new FieldVector3D<>(x.getField().getOne(), this, factor.negate(), v);
    }

    /** Subtract a scaled vector from the instance.
     * @param factor scale factor to apply to v before subtracting it
     * @param v vector to subtract
     * @return a new vector
     */
    public FieldVector3D<T> subtract(final T factor, final Vector3D v) {
        return new FieldVector3D<>(x.subtract(factor.multiply(v.getX())),
                y.subtract(factor.multiply(v.getY())),
                z.subtract(factor.multiply(v.getZ())));
    }

    /** Subtract a scaled vector from the instance.
     * @param factor scale factor to apply to v before subtracting it
     * @param v vector to subtract
     * @return a new vector
     */
    public FieldVector3D<T> subtract(final double factor, final FieldVector3D<T> v) {
        return new FieldVector3D<>(1.0, this, -factor, v);
    }

    /** Subtract a scaled vector from the instance.
     * @param factor scale factor to apply to v before subtracting it
     * @param v vector to subtract
     * @return a new vector
     */
    public FieldVector3D<T> subtract(final double factor, final Vector3D v) {
        return new FieldVector3D<>(x.subtract(factor * v.getX()),
                y.subtract(factor * v.getY()),
                z.subtract(factor * v.getZ()));
    }

    /** Get a normalized vector aligned with the instance.
     * @return a new normalized vector
     * @exception MathRuntimeException if the norm is zero
     */
    public FieldVector3D<T> normalize() throws MathRuntimeException {
        final T s = getNorm();
        if (s.getReal() == 0) {
            throw new MathRuntimeException(LocalizedGeometryFormats.CANNOT_NORMALIZE_A_ZERO_NORM_VECTOR);
        }
        return scalarMultiply(s.reciprocal());
    }

    /** Get a vector orthogonal to the instance.
     * <p>There are an infinite number of normalized vectors orthogonal
     * to the instance. This method picks up one of them almost
     * arbitrarily. It is useful when one needs to compute a reference
     * frame with one of the axes in a predefined direction. The
     * following example shows how to build a frame having the k axis
     * aligned with the known vector u :
     * </p>
     * <pre><code>
     *   Vector3D k = u.normalize();
     *   Vector3D i = k.orthogonal();
     *   Vector3D j = Vector3D.crossProduct(k, i);
     * </code></pre>
     * @return a new normalized vector orthogonal to the instance
     * @exception MathRuntimeException if the norm of the instance is null
     */
    public FieldVector3D<T> orthogonal() throws MathRuntimeException {

        final double threshold = 0.6 * getNorm().getReal();
        if (threshold == 0) {
            throw new MathRuntimeException(LocalizedCoreFormats.ZERO_NORM);
        }

        if (FastMath.abs(x.getReal()) <= threshold) {
            final T inverse  = y.square().add(z.square()).sqrt().reciprocal();
            return new FieldVector3D<>(inverse.getField().getZero(), inverse.multiply(z), inverse.multiply(y).negate());
        } else if (FastMath.abs(y.getReal()) <= threshold) {
            final T inverse  = x.square().add(z.square()).sqrt().reciprocal();
            return new FieldVector3D<>(inverse.multiply(z).negate(), inverse.getField().getZero(), inverse.multiply(x));
        } else {
            final T inverse  = x.square().add(y.square()).sqrt().reciprocal();
            return new FieldVector3D<>(inverse.multiply(y), inverse.multiply(x).negate(), inverse.getField().getZero());
        }

    }

    /** Compute the angular separation between two vectors.
     * <p>This method computes the angular separation between two
     * vectors using the dot product for well separated vectors and the
     * cross product for almost aligned vectors. This allows to have a
     * good accuracy in all cases, even for vectors very close to each
     * other.</p>
     * @param v1 first vector
     * @param v2 second vector
     * @param <T> the type of the field elements
     * @return angular separation between v1 and v2
     * @exception MathRuntimeException if either vector has a null norm
     */
    public static <T extends CalculusFieldElement<T>> T angle(final FieldVector3D<T> v1, final FieldVector3D<T> v2)
        throws MathRuntimeException {

        final T normProduct = v1.getNorm().multiply(v2.getNorm());
        if (normProduct.getReal() == 0) {
            throw new MathRuntimeException(LocalizedCoreFormats.ZERO_NORM);
        }

        final T dot = dotProduct(v1, v2);
        final double threshold = normProduct.getReal() * 0.9999;
        if ((dot.getReal() < -threshold) || (dot.getReal() > threshold)) {
            // the vectors are almost aligned, compute using the sine
            FieldVector3D<T> v3 = crossProduct(v1, v2);
            if (dot.getReal() >= 0) {
                return v3.getNorm().divide(normProduct).asin();
            }
            return v3.getNorm().divide(normProduct).asin().subtract(dot.getPi()).negate();
        }

        // the vectors are sufficiently separated to use the cosine
        return dot.divide(normProduct).acos();

    }

    /** Compute the angular separation between two vectors.
     * <p>This method computes the angular separation between two
     * vectors using the dot product for well separated vectors and the
     * cross product for almost aligned vectors. This allows to have a
     * good accuracy in all cases, even for vectors very close to each
     * other.</p>
     * @param v1 first vector
     * @param v2 second vector
     * @param <T> the type of the field elements
     * @return angular separation between v1 and v2
     * @exception MathRuntimeException if either vector has a null norm
     */
    public static <T extends CalculusFieldElement<T>> T angle(final FieldVector3D<T> v1, final Vector3D v2)
        throws MathRuntimeException {

        final T normProduct = v1.getNorm().multiply(v2.getNorm());
        if (normProduct.getReal() == 0) {
            throw new MathRuntimeException(LocalizedCoreFormats.ZERO_NORM);
        }

        final T dot = dotProduct(v1, v2);
        final double threshold = normProduct.getReal() * 0.9999;
        if ((dot.getReal() < -threshold) || (dot.getReal() > threshold)) {
            // the vectors are almost aligned, compute using the sine
            FieldVector3D<T> v3 = crossProduct(v1, v2);
            if (dot.getReal() >= 0) {
                return v3.getNorm().divide(normProduct).asin();
            }
            return v3.getNorm().divide(normProduct).asin().subtract(dot.getPi()).negate();
        }

        // the vectors are sufficiently separated to use the cosine
        return dot.divide(normProduct).acos();

    }

    /** Compute the angular separation between two vectors.
     * <p>This method computes the angular separation between two
     * vectors using the dot product for well separated vectors and the
     * cross product for almost aligned vectors. This allows to have a
     * good accuracy in all cases, even for vectors very close to each
     * other.</p>
     * @param v1 first vector
     * @param v2 second vector
     * @param <T> the type of the field elements
     * @return angular separation between v1 and v2
     * @exception MathRuntimeException if either vector has a null norm
     */
    public static <T extends CalculusFieldElement<T>> T angle(final Vector3D v1, final FieldVector3D<T> v2)
        throws MathRuntimeException {
        return angle(v2, v1);
    }

    /** Get the opposite of the instance.
     * @return a new vector which is opposite to the instance
     */
    public FieldVector3D<T> negate() {
        return new FieldVector3D<>(x.negate(), y.negate(), z.negate());
    }

    /** Multiply the instance by a scalar.
     * @param a scalar
     * @return a new vector
     */
    public FieldVector3D<T> scalarMultiply(final T a) {
        return new FieldVector3D<>(x.multiply(a), y.multiply(a), z.multiply(a));
    }

    /** Multiply the instance by a scalar.
     * @param a scalar
     * @return a new vector
     */
    public FieldVector3D<T> scalarMultiply(final double a) {
        return new FieldVector3D<>(x.multiply(a), y.multiply(a), z.multiply(a));
    }

    /**
     * Returns true if any coordinate of this vector is NaN; false otherwise
     * @return  true if any coordinate of this vector is NaN; false otherwise
     */
    public boolean isNaN() {
        return Double.isNaN(x.getReal()) || Double.isNaN(y.getReal()) || Double.isNaN(z.getReal());
    }

    /**
     * Returns true if any coordinate of this vector is infinite and none are NaN;
     * false otherwise
     * @return  true if any coordinate of this vector is infinite and none are NaN;
     * false otherwise
     */
    public boolean isInfinite() {
        return !isNaN() && (Double.isInfinite(x.getReal()) || Double.isInfinite(y.getReal()) || Double.isInfinite(z.getReal()));
    }

    /**
     * Test for the equality of two 3D vectors.
     * <p>
     * If all coordinates of two 3D vectors are exactly the same, and none of their
     * {@link CalculusFieldElement#getReal() real part} are <code>NaN</code>, the
     * two 3D vectors are considered to be equal.
     * </p>
     * <p>
     * <code>NaN</code> coordinates are considered to affect globally the vector
     * and be equals to each other - i.e, if either (or all) real part of the
     * coordinates of the 3D vector are <code>NaN</code>, the 3D vector is <code>NaN</code>.
     * </p>
     *
     * @param other Object to test for equality to this
     * @return true if two 3D vector objects are equal, false if
     *         object is null, not an instance of FieldVector3D, or
     *         not equal to this FieldVector3D instance
     *
     */
    @Override
    public boolean equals(Object other) {

        if (this == other) {
            return true;
        }

        if (other instanceof FieldVector3D) {
            @SuppressWarnings("unchecked")
            final FieldVector3D<T> rhs = (FieldVector3D<T>) other;
            if (rhs.isNaN()) {
                return this.isNaN();
            }

            return x.equals(rhs.x) && y.equals(rhs.y) && z.equals(rhs.z);

        }
        return false;
    }

    /**
     * Get a hashCode for the 3D vector.
     * <p>
     * All NaN values have the same hash code.</p>
     *
     * @return a hash code value for this object
     */
    @Override
    public int hashCode() {
        if (isNaN()) {
            return 409;
        }
        return 311 * (107 * x.hashCode() + 83 * y.hashCode() +  z.hashCode());
    }

    /** Compute the dot-product of the instance and another vector.
     * <p>
     * The implementation uses specific multiplication and addition
     * algorithms to preserve accuracy and reduce cancellation effects.
     * It should be very accurate even for nearly orthogonal vectors.
     * </p>
     * @see MathArrays#linearCombination(double, double, double, double, double, double)
     * @param v second vector
     * @return the dot product this.v
     */
    public T dotProduct(final FieldVector3D<T> v) {
        return x.linearCombination(x, v.x, y, v.y, z, v.z);
    }

    /** Compute the dot-product of the instance and another vector.
     * <p>
     * The implementation uses specific multiplication and addition
     * algorithms to preserve accuracy and reduce cancellation effects.
     * It should be very accurate even for nearly orthogonal vectors.
     * </p>
     * @see MathArrays#linearCombination(double, double, double, double, double, double)
     * @param v second vector
     * @return the dot product this.v
     */
    public T dotProduct(final Vector3D v) {
        return x.linearCombination(v.getX(), x, v.getY(), y, v.getZ(), z);
    }

    /** Compute the cross-product of the instance with another vector.
     * @param v other vector
     * @return the cross product this ^ v as a new Vector3D
     */
    public FieldVector3D<T> crossProduct(final FieldVector3D<T> v) {
        return new FieldVector3D<>(x.linearCombination(y, v.z, z.negate(), v.y),
                y.linearCombination(z, v.x, x.negate(), v.z),
                z.linearCombination(x, v.y, y.negate(), v.x));
    }

    /** Compute the cross-product of the instance with another vector.
     * @param v other vector
     * @return the cross product this ^ v as a new Vector3D
     */
    public FieldVector3D<T> crossProduct(final Vector3D v) {
        return new FieldVector3D<>(x.linearCombination(v.getZ(), y, -v.getY(), z),
                y.linearCombination(v.getX(), z, -v.getZ(), x),
                z.linearCombination(v.getY(), x, -v.getX(), y));
    }

    /** Compute the distance between the instance and another vector according to the L<sub>1</sub> norm.
     * <p>Calling this method is equivalent to calling:
     * <code>q.subtract(p).getNorm1()</code> except that no intermediate
     * vector is built</p>
     * @param v second vector
     * @return the distance between the instance and p according to the L<sub>1</sub> norm
     */
    public T distance1(final FieldVector3D<T> v) {
        final T dx = v.x.subtract(x).abs();
        final T dy = v.y.subtract(y).abs();
        final T dz = v.z.subtract(z).abs();
        return dx.add(dy).add(dz);
    }

    /** Compute the distance between the instance and another vector according to the L<sub>1</sub> norm.
     * <p>Calling this method is equivalent to calling:
     * <code>q.subtract(p).getNorm1()</code> except that no intermediate
     * vector is built</p>
     * @param v second vector
     * @return the distance between the instance and p according to the L<sub>1</sub> norm
     */
    public T distance1(final Vector3D v) {
        final T dx = x.subtract(v.getX()).abs();
        final T dy = y.subtract(v.getY()).abs();
        final T dz = z.subtract(v.getZ()).abs();
        return dx.add(dy).add(dz);
    }

    /** Compute the distance between the instance and another vector according to the L<sub>2</sub> norm.
     * <p>Calling this method is equivalent to calling:
     * <code>q.subtract(p).getNorm()</code> except that no intermediate
     * vector is built</p>
     * @param v second vector
     * @return the distance between the instance and p according to the L<sub>2</sub> norm
     */
    public T distance(final FieldVector3D<T> v) {
        final T dx = v.x.subtract(x);
        final T dy = v.y.subtract(y);
        final T dz = v.z.subtract(z);
        return dx.square().add(dy.square()).add(dz.square()).sqrt();
    }

    /** Compute the distance between the instance and another vector according to the L<sub>2</sub> norm.
     * <p>Calling this method is equivalent to calling:
     * <code>q.subtract(p).getNorm()</code> except that no intermediate
     * vector is built</p>
     * @param v second vector
     * @return the distance between the instance and p according to the L<sub>2</sub> norm
     */
    public T distance(final Vector3D v) {
        final T dx = x.subtract(v.getX());
        final T dy = y.subtract(v.getY());
        final T dz = z.subtract(v.getZ());
        return dx.square().add(dy.square()).add(dz.square()).sqrt();
    }

    /** Compute the distance between the instance and another vector according to the L<sub>&infin;</sub> norm.
     * <p>Calling this method is equivalent to calling:
     * <code>q.subtract(p).getNormInf()</code> except that no intermediate
     * vector is built</p>
     * @param v second vector
     * @return the distance between the instance and p according to the L<sub>&infin;</sub> norm
     */
    public T distanceInf(final FieldVector3D<T> v) {
        final T dx = v.x.subtract(x).abs();
        final T dy = v.y.subtract(y).abs();
        final T dz = v.z.subtract(z).abs();
        if (dx.getReal() <= dy.getReal()) {
            if (dy.getReal() <= dz.getReal()) {
                return dz;
            } else {
                return dy;
            }
        } else {
            if (dx.getReal() <= dz.getReal()) {
                return dz;
            } else {
                return dx;
            }
        }
    }

    /** Compute the distance between the instance and another vector according to the L<sub>&infin;</sub> norm.
     * <p>Calling this method is equivalent to calling:
     * <code>q.subtract(p).getNormInf()</code> except that no intermediate
     * vector is built</p>
     * @param v second vector
     * @return the distance between the instance and p according to the L<sub>&infin;</sub> norm
     */
    public T distanceInf(final Vector3D v) {
        final T dx = x.subtract(v.getX()).abs();
        final T dy = y.subtract(v.getY()).abs();
        final T dz = z.subtract(v.getZ()).abs();
        if (dx.getReal() <= dy.getReal()) {
            if (dy.getReal() <= dz.getReal()) {
                return dz;
            } else {
                return dy;
            }
        } else {
            if (dx.getReal() <= dz.getReal()) {
                return dz;
            } else {
                return dx;
            }
        }
    }

    /** Compute the square of the distance between the instance and another vector.
     * <p>Calling this method is equivalent to calling:
     * <code>q.subtract(p).getNormSq()</code> except that no intermediate
     * vector is built</p>
     * @param v second vector
     * @return the square of the distance between the instance and p
     */
    public T distanceSq(final FieldVector3D<T> v) {
        final T dx = v.x.subtract(x);
        final T dy = v.y.subtract(y);
        final T dz = v.z.subtract(z);
        return dx.square().add(dy.square()).add(dz.square());
    }

    /** Compute the square of the distance between the instance and another vector.
     * <p>Calling this method is equivalent to calling:
     * <code>q.subtract(p).getNormSq()</code> except that no intermediate
     * vector is built</p>
     * @param v second vector
     * @return the square of the distance between the instance and p
     */
    public T distanceSq(final Vector3D v) {
        final T dx = x.subtract(v.getX());
        final T dy = y.subtract(v.getY());
        final T dz = z.subtract(v.getZ());
        return dx.square().add(dy.square()).add(dz.square());
    }

    /** Compute the dot-product of two vectors.
     * @param v1 first vector
     * @param v2 second vector
     * @param <T> the type of the field elements
     * @return the dot product v1.v2
     */
    public static <T extends CalculusFieldElement<T>> T dotProduct(final FieldVector3D<T> v1,
                                                                   final FieldVector3D<T> v2) {
        return v1.dotProduct(v2);
    }

    /** Compute the dot-product of two vectors.
     * @param v1 first vector
     * @param v2 second vector
     * @param <T> the type of the field elements
     * @return the dot product v1.v2
     */
    public static <T extends CalculusFieldElement<T>> T dotProduct(final FieldVector3D<T> v1,
                                                                   final Vector3D v2) {
        return v1.dotProduct(v2);
    }

    /** Compute the dot-product of two vectors.
     * @param v1 first vector
     * @param v2 second vector
     * @param <T> the type of the field elements
     * @return the dot product v1.v2
     */
    public static <T extends CalculusFieldElement<T>> T dotProduct(final Vector3D v1,
                                                                   final FieldVector3D<T> v2) {
        return v2.dotProduct(v1);
    }

    /** Compute the cross-product of two vectors.
     * @param v1 first vector
     * @param v2 second vector
     * @param <T> the type of the field elements
     * @return the cross product v1 ^ v2 as a new Vector
     */
    public static <T extends CalculusFieldElement<T>> FieldVector3D<T> crossProduct(final FieldVector3D<T> v1,
                                                                                    final FieldVector3D<T> v2) {
        return v1.crossProduct(v2);
    }

    /** Compute the cross-product of two vectors.
     * @param v1 first vector
     * @param v2 second vector
     * @param <T> the type of the field elements
     * @return the cross product v1 ^ v2 as a new Vector
     */
    public static <T extends CalculusFieldElement<T>> FieldVector3D<T> crossProduct(final FieldVector3D<T> v1,
                                                                                    final Vector3D v2) {
        return v1.crossProduct(v2);
    }

    /** Compute the cross-product of two vectors.
     * @param v1 first vector
     * @param v2 second vector
     * @param <T> the type of the field elements
     * @return the cross product v1 ^ v2 as a new Vector
     */
    public static <T extends CalculusFieldElement<T>> FieldVector3D<T> crossProduct(final Vector3D v1,
                                                                                    final FieldVector3D<T> v2) {
        return new FieldVector3D<>(v2.x.linearCombination(v1.getY(), v2.z, -v1.getZ(), v2.y),
                v2.y.linearCombination(v1.getZ(), v2.x, -v1.getX(), v2.z),
                v2.z.linearCombination(v1.getX(), v2.y, -v1.getY(), v2.x));
    }

    /** Compute the distance between two vectors according to the L<sub>1</sub> norm.
     * <p>Calling this method is equivalent to calling:
     * <code>v1.subtract(v2).getNorm1()</code> except that no intermediate
     * vector is built</p>
     * @param v1 first vector
     * @param v2 second vector
     * @param <T> the type of the field elements
     * @return the distance between v1 and v2 according to the L<sub>1</sub> norm
     */
    public static <T extends CalculusFieldElement<T>> T distance1(final FieldVector3D<T> v1,
                                                                  final FieldVector3D<T> v2) {
        return v1.distance1(v2);
    }

    /** Compute the distance between two vectors according to the L<sub>1</sub> norm.
     * <p>Calling this method is equivalent to calling:
     * <code>v1.subtract(v2).getNorm1()</code> except that no intermediate
     * vector is built</p>
     * @param v1 first vector
     * @param v2 second vector
     * @param <T> the type of the field elements
     * @return the distance between v1 and v2 according to the L<sub>1</sub> norm
     */
    public static <T extends CalculusFieldElement<T>> T distance1(final FieldVector3D<T> v1,
                                                                  final Vector3D v2) {
        return v1.distance1(v2);
    }

    /** Compute the distance between two vectors according to the L<sub>1</sub> norm.
     * <p>Calling this method is equivalent to calling:
     * <code>v1.subtract(v2).getNorm1()</code> except that no intermediate
     * vector is built</p>
     * @param v1 first vector
     * @param v2 second vector
     * @param <T> the type of the field elements
     * @return the distance between v1 and v2 according to the L<sub>1</sub> norm
     */
    public static <T extends CalculusFieldElement<T>> T distance1(final Vector3D v1,
                                                                  final FieldVector3D<T> v2) {
        return v2.distance1(v1);
    }

    /** Compute the distance between two vectors according to the L<sub>2</sub> norm.
     * <p>Calling this method is equivalent to calling:
     * <code>v1.subtract(v2).getNorm()</code> except that no intermediate
     * vector is built</p>
     * @param v1 first vector
     * @param v2 second vector
     * @param <T> the type of the field elements
     * @return the distance between v1 and v2 according to the L<sub>2</sub> norm
     */
    public static <T extends CalculusFieldElement<T>> T distance(final FieldVector3D<T> v1,
                                                                 final FieldVector3D<T> v2) {
        return v1.distance(v2);
    }

    /** Compute the distance between two vectors according to the L<sub>2</sub> norm.
     * <p>Calling this method is equivalent to calling:
     * <code>v1.subtract(v2).getNorm()</code> except that no intermediate
     * vector is built</p>
     * @param v1 first vector
     * @param v2 second vector
     * @param <T> the type of the field elements
     * @return the distance between v1 and v2 according to the L<sub>2</sub> norm
     */
    public static <T extends CalculusFieldElement<T>> T distance(final FieldVector3D<T> v1,
                                                                 final Vector3D v2) {
        return v1.distance(v2);
    }

    /** Compute the distance between two vectors according to the L<sub>2</sub> norm.
     * <p>Calling this method is equivalent to calling:
     * <code>v1.subtract(v2).getNorm()</code> except that no intermediate
     * vector is built</p>
     * @param v1 first vector
     * @param v2 second vector
     * @param <T> the type of the field elements
     * @return the distance between v1 and v2 according to the L<sub>2</sub> norm
     */
    public static <T extends CalculusFieldElement<T>> T distance(final Vector3D v1,
                                                                 final FieldVector3D<T> v2) {
        return v2.distance(v1);
    }

    /** Compute the distance between two vectors according to the L<sub>&infin;</sub> norm.
     * <p>Calling this method is equivalent to calling:
     * <code>v1.subtract(v2).getNormInf()</code> except that no intermediate
     * vector is built</p>
     * @param v1 first vector
     * @param v2 second vector
     * @param <T> the type of the field elements
     * @return the distance between v1 and v2 according to the L<sub>&infin;</sub> norm
     */
    public static <T extends CalculusFieldElement<T>> T distanceInf(final FieldVector3D<T> v1,
                                                                    final FieldVector3D<T> v2) {
        return v1.distanceInf(v2);
    }

    /** Compute the distance between two vectors according to the L<sub>&infin;</sub> norm.
     * <p>Calling this method is equivalent to calling:
     * <code>v1.subtract(v2).getNormInf()</code> except that no intermediate
     * vector is built</p>
     * @param v1 first vector
     * @param v2 second vector
     * @param <T> the type of the field elements
     * @return the distance between v1 and v2 according to the L<sub>&infin;</sub> norm
     */
    public static <T extends CalculusFieldElement<T>> T distanceInf(final FieldVector3D<T> v1,
                                                                    final Vector3D v2) {
        return v1.distanceInf(v2);
    }

    /** Compute the distance between two vectors according to the L<sub>&infin;</sub> norm.
     * <p>Calling this method is equivalent to calling:
     * <code>v1.subtract(v2).getNormInf()</code> except that no intermediate
     * vector is built</p>
     * @param v1 first vector
     * @param v2 second vector
     * @param <T> the type of the field elements
     * @return the distance between v1 and v2 according to the L<sub>&infin;</sub> norm
     */
    public static <T extends CalculusFieldElement<T>> T distanceInf(final Vector3D v1,
                                                                    final FieldVector3D<T> v2) {
        return v2.distanceInf(v1);
    }

    /** Compute the square of the distance between two vectors.
     * <p>Calling this method is equivalent to calling:
     * <code>v1.subtract(v2).getNormSq()</code> except that no intermediate
     * vector is built</p>
     * @param v1 first vector
     * @param v2 second vector
     * @param <T> the type of the field elements
     * @return the square of the distance between v1 and v2
     */
    public static <T extends CalculusFieldElement<T>> T distanceSq(final FieldVector3D<T> v1,
                                                                   final FieldVector3D<T> v2) {
        return v1.distanceSq(v2);
    }

    /** Compute the square of the distance between two vectors.
     * <p>Calling this method is equivalent to calling:
     * <code>v1.subtract(v2).getNormSq()</code> except that no intermediate
     * vector is built</p>
     * @param v1 first vector
     * @param v2 second vector
     * @param <T> the type of the field elements
     * @return the square of the distance between v1 and v2
     */
    public static <T extends CalculusFieldElement<T>> T distanceSq(final FieldVector3D<T> v1,
                                                                   final Vector3D v2) {
        return v1.distanceSq(v2);
    }

    /** Compute the square of the distance between two vectors.
     * <p>Calling this method is equivalent to calling:
     * <code>v1.subtract(v2).getNormSq()</code> except that no intermediate
     * vector is built</p>
     * @param v1 first vector
     * @param v2 second vector
     * @param <T> the type of the field elements
     * @return the square of the distance between v1 and v2
     */
    public static <T extends CalculusFieldElement<T>> T distanceSq(final Vector3D v1,
                                                                   final FieldVector3D<T> v2) {
        return v2.distanceSq(v1);
    }

    /** Get a string representation of this vector.
     * @return a string representation of this vector
     */
    @Override
    public String toString() {
        return Vector3DFormat.getVector3DFormat().format(toVector3D());
    }

    /** Get a string representation of this vector.
     * @param format the custom format for components
     * @return a string representation of this vector
     */
    public String toString(final NumberFormat format) {
        return new Vector3DFormat(format).format(toVector3D());
    }

    /** {@inheritDoc} */
    @Override
    public FieldVector3D<T> blendArithmeticallyWith(final FieldVector3D<T> other, final T blendingValue)
            throws MathIllegalArgumentException {
        SmoothStepFactory.checkBetweenZeroAndOneIncluded(blendingValue.getReal());
        final T one = x.getField().getOne();
        return this.scalarMultiply(one.subtract(blendingValue)).add(other.scalarMultiply(blendingValue));
    }
}