CS426 Computer Graphics - Homogeneous Coordinates

Flashcards covering homogeneous coordinates, transformations, and frames in 3D computer graphics.

Euclidean transformation

Rigid motions (translation and rotation), 6 degrees of freedom.

Similarity transformation

Rigid motions and uniform scaling (translation and rotation), 7 degrees of freedom.

Affine transformation

Linear mappings (shear), scaling, and rigid motions, 12 degrees of freedom.

Projective transformation -homography

Maps lines to lines, change of view, 15 degrees of freedom.

Homogeneous Coordinates

x, y, z, w where w is a scaling factor, typically 1. Represented as 𝑣 = 𝑎𝑖 + 𝑏𝑗 + 𝑐𝑘 where i, j, and k are unit vectors along the X, Y, and Z axes.

Direction Vector

When w=0 the vector (whatever the values of a,b,c) is positioned at infinity. Carries direction information only.

Position Vector Conversion

To convert a position vector into a direction vector to find the unit vector that points in the same direction.

Points vs. Directions in Homogeneous Coordinates

Points: 𝑥 𝑦 𝑧 𝑤 ≠ 0 ; Directions: 𝑥 𝑦 𝑧 𝑤 = 0.

Frame

Three mutually perpendicular vectors (n, o, a) centered at a point (Px, Py, Pz) in a reference frame.

Components of a Frame

First three vectors (n, o, a) define the orientation, and the fourth is used for position/translation.

Cross Product for Finding Vectors

Use cross product to find the vector at right angles to two others. a = n x o

Properties of Frame Vectors

The dot product of any two (a, o, n) vectors is zero, and the length of any one vector is one.

Translation in 3D

Requires a 4x4 matrix to translate a 3D (x,y,z) point represented as (Px +dx,Py+dy, Pz+dz).

Representing a Rigid Body

Attaching a frame to it, following the motion of the frame translates to the motion of the object.

Final Transformation Matrix Order

Matrix.CreateScale(scale) * Matrix.CreateRotationX(roll) * Matrix.CreateRotationY(pitch) * Matrix.CreateRotationZ(yaw) * Matrix.CreateTranslation(newVector(dx, dy, dz)).