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diff --git a/src/main/java/org/apache/commons/math3/geometry/spherical/twod/S2Point.java b/src/main/java/org/apache/commons/math3/geometry/spherical/twod/S2Point.java
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+/*
+ * 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
+ *
+ *      http://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.
+ */
+package org.apache.commons.math3.geometry.spherical.twod;
+
+import org.apache.commons.math3.exception.MathArithmeticException;
+import org.apache.commons.math3.exception.OutOfRangeException;
+import org.apache.commons.math3.geometry.Point;
+import org.apache.commons.math3.geometry.Space;
+import org.apache.commons.math3.geometry.euclidean.threed.Vector3D;
+import org.apache.commons.math3.util.FastMath;
+import org.apache.commons.math3.util.MathUtils;
+
+/** This class represents a point on the 2-sphere.
+ * <p>
+ * We use the mathematical convention to use the azimuthal angle \( \theta \)
+ * in the x-y plane as the first coordinate, and the polar angle \( \varphi \)
+ * as the second coordinate (see <a
+ * href="http://mathworld.wolfram.com/SphericalCoordinates.html">Spherical
+ * Coordinates</a> in MathWorld).
+ * </p>
+ * <p>Instances of this class are guaranteed to be immutable.</p>
+ * @since 3.3
+ */
+public class S2Point implements Point<Sphere2D> {
+
+    /** +I (coordinates: \( \theta = 0, \varphi = \pi/2 \)). */
+    public static final S2Point PLUS_I = new S2Point(0, 0.5 * FastMath.PI, Vector3D.PLUS_I);
+
+    /** +J (coordinates: \( \theta = \pi/2, \varphi = \pi/2 \))). */
+    public static final S2Point PLUS_J = new S2Point(0.5 * FastMath.PI, 0.5 * FastMath.PI, Vector3D.PLUS_J);
+
+    /** +K (coordinates: \( \theta = any angle, \varphi = 0 \)). */
+    public static final S2Point PLUS_K = new S2Point(0, 0, Vector3D.PLUS_K);
+
+    /** -I (coordinates: \( \theta = \pi, \varphi = \pi/2 \)). */
+    public static final S2Point MINUS_I = new S2Point(FastMath.PI, 0.5 * FastMath.PI, Vector3D.MINUS_I);
+
+    /** -J (coordinates: \( \theta = 3\pi/2, \varphi = \pi/2 \)). */
+    public static final S2Point MINUS_J = new S2Point(1.5 * FastMath.PI, 0.5 * FastMath.PI, Vector3D.MINUS_J);
+
+    /** -K (coordinates: \( \theta = any angle, \varphi = \pi \)). */
+    public static final S2Point MINUS_K = new S2Point(0, FastMath.PI, Vector3D.MINUS_K);
+
+    // CHECKSTYLE: stop ConstantName
+    /** A vector with all coordinates set to NaN. */
+    public static final S2Point NaN = new S2Point(Double.NaN, Double.NaN, Vector3D.NaN);
+    // CHECKSTYLE: resume ConstantName
+
+    /** Serializable UID. */
+    private static final long serialVersionUID = 20131218L;
+
+    /** Azimuthal angle \( \theta \) in the x-y plane. */
+    private final double theta;
+
+    /** Polar angle \( \varphi \). */
+    private final double phi;
+
+    /** Corresponding 3D normalized vector. */
+    private final Vector3D vector;
+
+    /** Simple constructor.
+     * Build a vector from its spherical coordinates
+     * @param theta azimuthal angle \( \theta \) in the x-y plane
+     * @param phi polar angle \( \varphi \)
+     * @see #getTheta()
+     * @see #getPhi()
+     * @exception OutOfRangeException if \( \varphi \) is not in the [\( 0; \pi \)] range
+     */
+    public S2Point(final double theta, final double phi)
+        throws OutOfRangeException {
+        this(theta, phi, vector(theta, phi));
+    }
+
+    /** Simple constructor.
+     * Build a vector from its underlying 3D vector
+     * @param vector 3D vector
+     * @exception MathArithmeticException if vector norm is zero
+     */
+    public S2Point(final Vector3D vector) throws MathArithmeticException {
+        this(FastMath.atan2(vector.getY(), vector.getX()), Vector3D.angle(Vector3D.PLUS_K, vector),
+             vector.normalize());
+    }
+
+    /** Build a point from its internal components.
+     * @param theta azimuthal angle \( \theta \) in the x-y plane
+     * @param phi polar angle \( \varphi \)
+     * @param vector corresponding vector
+     */
+    private S2Point(final double theta, final double phi, final Vector3D vector) {
+        this.theta  = theta;
+        this.phi    = phi;
+        this.vector = vector;
+    }
+
+    /** Build the normalized vector corresponding to spherical coordinates.
+     * @param theta azimuthal angle \( \theta \) in the x-y plane
+     * @param phi polar angle \( \varphi \)
+     * @return normalized vector
+     * @exception OutOfRangeException if \( \varphi \) is not in the [\( 0; \pi \)] range
+     */
+    private static Vector3D vector(final double theta, final double phi)
+       throws OutOfRangeException {
+
+        if (phi < 0 || phi > FastMath.PI) {
+            throw new OutOfRangeException(phi, 0, FastMath.PI);
+        }
+
+        final double cosTheta = FastMath.cos(theta);
+        final double sinTheta = FastMath.sin(theta);
+        final double cosPhi   = FastMath.cos(phi);
+        final double sinPhi   = FastMath.sin(phi);
+
+        return new Vector3D(cosTheta * sinPhi, sinTheta * sinPhi, cosPhi);
+
+    }
+
+    /** Get the azimuthal angle \( \theta \) in the x-y plane.
+     * @return azimuthal angle \( \theta \) in the x-y plane
+     * @see #S2Point(double, double)
+     */
+    public double getTheta() {
+        return theta;
+    }
+
+    /** Get the polar angle \( \varphi \).
+     * @return polar angle \( \varphi \)
+     * @see #S2Point(double, double)
+     */
+    public double getPhi() {
+        return phi;
+    }
+
+    /** Get the corresponding normalized vector in the 3D euclidean space.
+     * @return normalized vector
+     */
+    public Vector3D getVector() {
+        return vector;
+    }
+
+    /** {@inheritDoc} */
+    public Space getSpace() {
+        return Sphere2D.getInstance();
+    }
+
+    /** {@inheritDoc} */
+    public boolean isNaN() {
+        return Double.isNaN(theta) || Double.isNaN(phi);
+    }
+
+    /** Get the opposite of the instance.
+     * @return a new vector which is opposite to the instance
+     */
+    public S2Point negate() {
+        return new S2Point(-theta, FastMath.PI - phi, vector.negate());
+    }
+
+    /** {@inheritDoc} */
+    public double distance(final Point<Sphere2D> point) {
+        return distance(this, (S2Point) point);
+    }
+
+    /** Compute the distance (angular separation) between two points.
+     * @param p1 first vector
+     * @param p2 second vector
+     * @return the angular separation between p1 and p2
+     */
+    public static double distance(S2Point p1, S2Point p2) {
+        return Vector3D.angle(p1.vector, p2.vector);
+    }
+
+    /**
+     * Test for the equality of two points on the 2-sphere.
+     * <p>
+     * If all coordinates of two points are exactly the same, and none are
+     * <code>Double.NaN</code>, the two points 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) coordinates of the
+     * 2D vector are equal to <code>Double.NaN</code>, the 2D vector is equal to
+     * {@link #NaN}.
+     * </p>
+     *
+     * @param other Object to test for equality to this
+     * @return true if two points on the 2-sphere objects are equal, false if
+     *         object is null, not an instance of S2Point, or
+     *         not equal to this S2Point instance
+     *
+     */
+    @Override
+    public boolean equals(Object other) {
+
+        if (this == other) {
+            return true;
+        }
+
+        if (other instanceof S2Point) {
+            final S2Point rhs = (S2Point) other;
+            if (rhs.isNaN()) {
+                return this.isNaN();
+            }
+
+            return (theta == rhs.theta) && (phi == rhs.phi);
+        }
+        return false;
+    }
+
+    /**
+     * Get a hashCode for the 2D 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 542;
+        }
+        return 134 * (37 * MathUtils.hash(theta) +  MathUtils.hash(phi));
+    }
+
+}