1 /*
2 * Licensed to the Apache Software Foundation (ASF) under one or more
3 * contributor license agreements. See the NOTICE file distributed with
4 * this work for additional information regarding copyright ownership.
5 * The ASF licenses this file to You under the Apache License, Version 2.0
6 * (the "License"); you may not use this file except in compliance with
7 * the License. You may obtain a copy of the License at
8 *
9 * http://www.apache.org/licenses/LICENSE-2.0
10 *
11 * Unless required by applicable law or agreed to in writing, software
12 * distributed under the License is distributed on an "AS IS" BASIS,
13 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
14 * See the License for the specific language governing permissions and
15 * limitations under the License.
16 */
17 package org.apache.commons.geometry.core;
18
19 import java.util.function.UnaryOperator;
20
21 /** Interface representing geometric transforms in a space, i.e. mappings from points to points.
22 * Implementations <em>must</em> fulfill a set of requirements, listed below, that preserve the
23 * consistency of partitionings on the space. Transforms that do not meet these requirements, while
24 * potentially valid mathematically, cannot be expected to produce correct results with algorithms
25 * that use this interface.
26 *
27 * <ol>
28 * <li>Transforms must represent functions that are <em>one-to-one</em> and <em>onto</em> (i.e.
29 * <a href="https://en.wikipedia.org/wiki/Bijection">bijections</a>). This means that every point
30 * in the space must be mapped to exactly one other point in the space. This also implies that the
31 * function is invertible.</li>
32 * <li>Transforms must preserve <a href="https://en.wikipedia.org/wiki/Collinearity">collinearity</a>.
33 * This means that if a set of points lie on a common hyperplane before the transform, then they must
34 * also lie on a common hyperplane after the transform. For example, if the Euclidean 2D points {@code a},
35 * {@code b}, and {@code c} lie on line {@code L}, then the transformed points {@code a'}, {@code b'}, and
36 * {@code c'} must lie on line {@code L'}, where {@code L'} is the transformed form of the line.</li>
37 * <li>Transforms must preserve the concept of
38 * <a href="https://en.wikipedia.org/wiki/Parallel_(geometry)">parallelism</a> defined for the space.
39 * This means that hyperplanes that are parallel before the transformation must remain parallel afterwards,
40 * and hyperplanes that intersect must also intersect afterwards. For example, a transform that causes parallel
41 * lines to converge to a single point in Euclidean space (such as the projective transforms used to create
42 * perspective viewpoints in 3D graphics) would not meet this requirement. However, a transform that turns
43 * a square into a rhombus with no right angles would fulfill the requirement, since the two pairs of parallel
44 * lines forming the square remain parallel after the transformation.
45 * </li>
46 * </ol>
47 *
48 * <p>Transforms that meet the above requirements in Euclidean space (and other affine spaces) are known as
49 * <a href="https://en.wikipedia.org/wiki/Affine_transformation">affine transforms</a>. Common affine transforms
50 * include translation, scaling, rotation, reflection, and any compositions thereof.
51 * </p>
52 *
53 * @param <P> Point implementation type
54 * @see <a href="https://en.wikipedia.org/wiki/Geometric_transformation">Geometric Transformation</a>
55 */
56 public interface Transform<P extends Point<P>> extends UnaryOperator<P> {
57
58 /** Get an instance representing the inverse transform.
59 * @return an instance representing the inverse transform
60 */
61 Transform<P> inverse();
62
63 /** Return true if the transform preserves the orientation of the space.
64 * For example, in Euclidean 2D space, this will be true for translations,
65 * rotations, and scalings but will be false for reflections.
66 * @return true if the transform preserves the orientation of the space
67 * @see <a href="https://en.wikipedia.org/wiki/Orientation_(vector_space)">Orientation</a>
68 */
69 boolean preservesOrientation();
70 }