Matrix Operations with Applications in Computer Graphics

Transcription

1 Matrix Operations with Applications in Computer Graphics

2 The zero matrix is a matrix with all zero entries. The identity matrix is the matrix I with on the main diagonal entries and for all other entries. x Special Matrices A diagonal matrix D [d ij ] has the property that d ij when i not equal to j. Sometimes we use the notation D diag{a,b,c}.

3 Matrix Multiplication Examples 4 A B 8 9 B A P Q 5 Q P [ ] N M [ ] M N M N

4 A matrix M : R -> R is called a linear transformation and maps vectors to vectors by YMX. The term linearity refers to the property that M(cU +V) cmu + MV for any scalar c and any vectors U and V Linear Transformation

5 The transpose of a matrix M [m ij ] is the matrix M T [m ji ]. That is the rows and columns are interchanged in M T (or the matrix is flipped about its main diagonal). T Matrix Transpose

6 A matrix M is symmetric if MM T. A matrix M is skew-symmetric if M T -M. T T Skew (Anti-) Symmetric

7 Computer Graphics Computer graphics is a study of the use of a computer to create and manipulate images and animated scenes, usually represented in three dimensions. A mathematical model of a three dimensional world uses vectors to describe the locations and material properties of objects and their relationships. An observer's location and line of sight are used to generate a perspective view of this mathematical model.

8 Scaling If a diagonal matrix D diag{d, d, d } has all positive entries, it is a scaling matrix. Each diagonal term represents how much stretching or shrinking occurs for the corresponding coordinate direction. Uniform scaling is D si diag{s,s,s} for s>.

9 A matrix is said to be invertible if there exists a matrix, M - such that MM - M - M I. An Invertible Matrix

10 Rotation A matrix R is a rotation matrix if its transpose and inverse are the same matrix, that is, R - R T, in which case RR T R T R I. The matrix has a corresponding unit-length axis of rotation U and angle of rotation φ. The choice is not unique since -U is also an axis of rotation and φ + πk for any integer k is an angle of rotation. If U(u,u,u ), we can define the skew-symmetric matrix S by S u u u u u u The rotation corresponding to axis U and angle φ is R I + (sin φ) S + ( - cos φ) S

11 Translation Translation of vectors by a fixed vector T element of R is represented by the function Y X + T for X and Y elements of R. It is not possible to represent this translation as a linear transformation of the form Y MX for some constant matrix M. However, if the problem is embedded in a fourdimensional space, it is possible to represent translation with a linear transformation (called a homogeneous transformation). ref: D Game Engine Design, by David H. Eberly, Morgan Kaufmann

12 Vector Model Orthographic Projection Shutterbug: Copyright 99 Pixar - Rendered by Thomas Williams and H. B. Siegel using Pixar's RenderMan TM

13 Perspective Projection Shutterbug: Copyright 99 Pixar - Rendered by Thomas Williams and H. B. Siegel using Pixar's RenderMan TM

14 Depth Cueing Shutterbug: Copyright 99 Pixar - Rendered by Thomas Williams and H. B. Siegel using Pixar's RenderMan TM

15 Depth Clipping Shutterbug: Copyright 99 Pixar - Rendered by Thomas Williams and H. B. Siegel using Pixar's RenderMan TM

16 Color Vectors Shutterbug: Copyright 99 Pixar - Rendered by Thomas Williams and H. B. Siegel using Pixar's RenderMan TM

17 Visible Line Determination Shutterbug: Copyright 99 Pixar - Rendered by Thomas Williams and H. B. Siegel using Pixar's RenderMan TM

18 Ambient Illumination Only Shutterbug: Copyright 99 Pixar - Rendered by Thomas Williams and H. B. Siegel using Pixar's RenderMan TM

19 Polygon Shading Shutterbug: Copyright 99 Pixar - Rendered by Thomas Williams and H. B. Siegel using Pixar's RenderMan TM

20 Gouraud Shading Shutterbug: Copyright 99 Pixar - Rendered by Thomas Williams and H. B. Siegel using Pixar's RenderMan TM

21 Gouraud Shading with Specular Reflection Shutterbug: Copyright 99 Pixar - Rendered by Thomas Williams and H. B. Siegel using Pixar's RenderMan TM

22 Phong Shading with Specular Reflection Shutterbug: Copyright 99 Pixar - Rendered by Thomas Williams and H. B. Siegel using Pixar's RenderMan TM

23 Curved Surfaces Shutterbug: Copyright 99 Pixar - Rendered by Thomas Williams and H. B. Siegel using Pixar's RenderMan TM

24 Local Lighing Sources Shutterbug: Copyright 99 Pixar - Rendered by Thomas Williams and H. B. Siegel using Pixar's RenderMan TM

25 Texture Mapping Shutterbug: Copyright 99 Pixar - Rendered by Thomas Williams and H. B. Siegel using Pixar's RenderMan TM

26 Displacement Mapping Shutterbug: Copyright 99 Pixar - Rendered by Thomas Williams and H. B. Siegel using Pixar's RenderMan TM

27 Reflection Mapping Shutterbug: Copyright 99 Pixar - Rendered by Thomas Williams and H. B. Siegel using Pixar's RenderMan TM

28 Homogeneous Transformations A vector (x,y,z) ε R can be mapped uniquely onto a vector (x,y,z,) ε R 4. vectors (x,y,z,w) ε R 4 can be projected onto the hyperplane w by Other (x,y,z,w) -> (x/w,y/w,z/w,). An entire line of points with with origin (,,,) is projected onto the single point (x,y,z,) All of R 4 \ {} is partitioned into equivalence classes, each class having representative projection (x,y,z,). A 4-tuple in this setting is called a homogeneous coordinate. Two homogeneous coordinates that are equivalent are indicated to be so by (x,y,z,w )~(x,y,z,w ).

29 Transformations can be applied to homogeneous coordinates to obtain other homogeneous coordinates. Such a 4x4 matrix H [h ij ], < i < and <j<, is called a homogeneous transformation as long as h. Usually, homogeneous matrices are written as x block matrices, where M is a x matrix, T is x S T is x and is a scalar. The product of a homogeneous coordinate and a homogeneous transformation in block format is, S T M H T + + w S wt MV w V S wt MV w V S T M H T T T ~

30 Any x linear transformation M can be represented by the homogeneous matrix Translation by a vector T can also be represented by a homogeneous transformation, The two transformations can be composed to represent Y MX + T as Assuming M is invertible, the equation can be solved for X M - (Y-T). Thus, the inverse of a homogeneous matrix is Perspective Transformations M T M X T M Y T M M T M

31 Perspective projection can also be represented by a homogenoeous matrix where the lower-left entry is not the zero vector. We usually discuss the geometric pipeline in terms of products of homogeneous transformations. That notation is a convenience and is not particularly useful in an implementation unless the underlying hardware (and/or graphics package) has native support for vector and matrix operations in four dimensions (e.g. opengl and SGI).