Transformation

Overview

A geometric transformation is a function that maps points in a geometric space to other points within the same space. These transformations can be represented in various forms, such as vector equations or matrix operations. Transformations are fundamental in geometry as they allow for the manipulation of objects in terms of their position, size, and shape. They can be generalized into different types, including linear, affine, and projective transformations, each possessing unique properties and capabilities.

In projective space, transformations are often represented as matrix multiplications. This representation is particularly powerful as it allows for the composition of multiple operations, such as translation, rotation, and scaling, into a single operation. This capability is illustrated in Figure 4.19, where a composition of transformations is consolidated into one multiplication instead of three.

Figure 4.19 This composition could be made into a single transformation using one multiplication instead of three.

Types of Transformations

Translation

Translation is a transformation that shifts points in space by adding specific coefficients to their coordinates. This operation effectively moves the entire space without altering the shape or orientation of the objects within it. For example, a point can be translated by adding dx to its x-coordinate and dy to its y-coordinate, resulting in new coordinates (xt, yt) where xt = xi + dx and yt = yi + dy.

Scaling

Scaling is a transformation that changes the size of objects by stretching or compressing the space along the axes. The degree of scaling is determined by specific coefficients, which can enlarge or shrink the objects. This transformation is crucial in applications where proportional resizing is required.

Rotation

Rotation is a transformation that pivots the space around a specific point, typically the origin, using trigonometric functions such as sine and cosine. This operation changes the orientation of objects without altering their shape or size. An example of a rotation transformation is rotating a rectangle 90 degrees clockwise around its center, which can be achieved through a composition of translation and rotation matrices.

Polynomial Transformations

Polynomial transformations involve using polynomials to model changes in position, size, or shape of objects. These transformations allow for complex manipulations across different dimensions and degrees, such as bilinear or biquadratic transformations. Polynomial transformations are particularly useful in advanced geometric modeling and computer graphics, where intricate transformations are required.