Geometric Transformations

Transformations are used in mathematics, computer graphics and visual communications. Transformations are also known as transits, slices, and cubes. A transformation is a mathematical process that transforms a polygon or any two dimensional objects on a flat surface or coordinate system into another form. Transformations describe the movement of two dimensional objects over a flat surface or coordinate system without reference to distance.


Transforms may be performed on a two dimensional surface with the Cartesian coordinate system, or on a point reference. A transform function takes an object t and performs a transform on it according to some specification. Some common transform functions are the translation, scaling, shearing and rotation of geometric units.

Transforms are mathematically defined as a set of transformation equations between any source and a target, performed on any coordinate frame. The coordinates of the source are taken in the x, y format, starting from zero (both horizontal and vertical) to the coordinate index that represents the origin of a point. Transforms may be real or complex. Real transforms simplify complex expressions by mapping each transformation to a scalar value, thus reducing the multiplication by an unknown factor. Complex transforms are more difficult to analyze mathematically.

A simple transform is the translation along a single axis. Let’s assume we are defining a figure whose origin is at (x, y) (coordinate system) and whose distance from the origin is given by c. A transform of this sort is simply the transformation of the point estimate c’ by any appropriate formula involving the coordinates. An example of a simple transform is the translation between x and y when expressed as a function of t. This function is also the basis of some of the more advanced transformation math such as the identity transform or the copy transform. An even simpler transformation is the translation between x and y, where t is a point within the x-axis defined by the origin and the x-axis direction. This is also an example of a complex transform.

There are many types of geometric transformations. One of the most important is the translation. The translation transforms the coordinate system from a source point P to a target point Q. The translation is based on the geometric translation of the geometric model of the target points. This involves translating each point in the model into a different coordinate system, then translating each point back to its original coordinate system at the destination. The main advantage of a geometric transformation is that it doesn’t have to be performed on every possible projection of the target shape, thus saving computational complexity.

Another common geometric transformation is the transformation of the translation of an image from a source point A to a target point B. The translation consists of translating each point in the figure into a separate coordinate plane. This is done by first finding the target shape in the image and then locating every point on the target shape in the source image that is tangent to the target shape. By translating each point in the figure (the t-interval) into its tangent coordinate plane, the transform is applied.

The pre-image transform is a simple but important transformation. It takes the original image, and changes it into a stencil that when viewed, creates the desired reflection of the target shape. This pre-image transform is based on the Ray-tracing transformation and is used widely in CAD-CAM based design systems.

The reflection transform is also important in reflecting light on any surface. Translate the image from a source point A to a target point B by defining the tangent planes of the triangles on the target surface. Then translate each triangle back onto its original plane. The reflection is then achieved when all of the triangles have their planes tangential to the target surface.

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