Applications of Computer and Video Graphics within Linear Algebra

By: Nicole Hill, Emily Deane & Bryan Combs Linear Algebra- the backbone of many applications from computer graphics to gaming

Projections

Squeezing

Shear Mapping

Scaling

Rotations

Reflections Projections

Linear transformations from a vector space to itself

Leaves the image unchanged

Two Types of graphical projections: Parallel Projections & Perspective Projections Parallel Projections

line of sight formations from the object to the projection plane which are parallel with one another

pictorials-show an image as an object viewed from a skew directions in order to reveal all three axes of space

Perspective Projections

Lines emerging from a single point

A 2D projection being viewed as though through a camera lens

Gives the effect of distant obects appearing smaller Linear Mapping

a mapping defined from one vector space to another that is linear

operations of vector addition and scalar multiplication Shear Mapping (also called transvection)

All the points move parallel to a fixed line or plane, in such a way that the distance a point moves is proportional to its distance from that line or plane Scaling

a linear transformation which enlarges or shrinks an object by a scale factor Rotations

linear transformations represented by orthogonal matrices with determinant 1

Since linear transformations are determined by their action on a basis, we will look at the standard basis in R2. The vectors <1,0> and <0,1> when rotated, arrive at new coordinates described by Reflections(may also be described as the product of two reflections)

Reflections across an axis that passes through the origin are linear transformations

Consists of finding the distance from each point to the axis (line in 2D, plane in 3D), and mapping it to a corresponding point the same distance from the axis on the opposite side. The result, interpreted visually, is a “mirror image” of the object being reflected about the axis.

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# Computer Graphics-Linear Algebra

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