Math for Computer Graphics and Computer Vision
From Math Images
(Difference between revisions)
(18 intermediate revisions not shown.)  
Line 1:  Line 1:  
Not surprisingly, the mathematics used in computer graphics was touched upon by the students at Drexel and Swarthmore during the summer of '09. Some of the [http://mathforum.org/mathimages/index.php/Helper_Pages Helper Pages], in particular are needed topics for some of the image pages, and for computer graphics.  Not surprisingly, the mathematics used in computer graphics was touched upon by the students at Drexel and Swarthmore during the summer of '09. Some of the [http://mathforum.org/mathimages/index.php/Helper_Pages Helper Pages], in particular are needed topics for some of the image pages, and for computer graphics.  
+  
As students turned their creative talents on these topics we decided to open this repository of software devoted to understanding them. Our hope is that it will develop into a useful resource for students and faculty in both disciplines. Please contribute your good material!  As students turned their creative talents on these topics we decided to open this repository of software devoted to understanding them. Our hope is that it will develop into a useful resource for students and faculty in both disciplines. Please contribute your good material!  
  
  :* Vectors and  +  The following is a list of mathematical topics used in computer graphics. The original list was provided by Drexel professor David Breen. 
+  
+  :* 2D, 3D, 4D real spaces; affine subspaces; homogeneous coordinates  
+  :* [[Vector Vectors]] and [[Matrix Matrices]]  
+  :* Graphics primitives  
+  ::* 2D primitives developed from triangles: fans, strips  
+  ::* Convexity and convex sums  
+  ::* 3D models based on 2D faces  
+  ::* ''Do we save machine representations for the CG course?''  
+  
:* Transformations  :* Transformations  
  :* Quaternions  +  ::*[[Transformations and Matrices]] 
+  :::* Primitive geometric transformations  
+  :::* Creating general transformations via sequences of primitives  
+  :::* Inverse transformations via primitives  
+  :::* ''Do we include transformation stacks?''  
+  ::*[[Change Of Coordinate Transformations]]  
+  :::* This might well be phrased in terms of the viewing transformation  
+  
+  :* [[QuaternionQuaternions]]  
+  
:* Hierarchical coordinate systems  :* Hierarchical coordinate systems  
+  
:* Geometry  :* Geometry  
  ::* Curves (CatmullRom, Bezier, Bspline)  +  ::* Implicit geometry  lines, circles, ellipses 
+  ::* [[Implicit Surfaces]]  quadrics, superquadrics  
+  ::* [[Implicit Equations]]  
+  ::* [[Parametric EquationsParametric geometry]]  lines, circles, ellipses  
+  ::* Curves (CatmullRom, [[Bezier CurvesBezier]], Bspline)  
+  ::* Parametric surfaces  quadrics, superquadrics, others  
::* Bezier patches  ::* Bezier patches  
::* Subdivision surfaces  ::* Subdivision surfaces  
  ::*  +  ::*[[Procedural Image]] 
  +  :* [[Surface Normals]]  
  :*  +  ::* Techniques of computing them from analytic and nonanalytic cases 
  :* Silhouette  +  :* [[Silhouette Edges]] 
:* Procedural texture maps  :* Procedural texture maps  
+  ::* Noise  
:* Rayobject intersection  :* Rayobject intersection  
  :* Perspective and parallel planar projections  +  ::* [[Bounding Volumes]]  Bounding spheres and boxes 
+  :* [[Planar ProjectionPerspective and parallel planar projections]]  
:* Nonplanar projections  :* Nonplanar projections  
:* Edge detection  :* Edge detection  
:* Fourier analysis and convolution  :* Fourier analysis and convolution  
+  
More examples may be found in the lecture slides of [http://www.cs.drexel.edu/~david/Classes/CS430 CS 430].  More examples may be found in the lecture slides of [http://www.cs.drexel.edu/~david/Classes/CS430 CS 430]. 
Current revision
Not surprisingly, the mathematics used in computer graphics was touched upon by the students at Drexel and Swarthmore during the summer of '09. Some of the Helper Pages, in particular are needed topics for some of the image pages, and for computer graphics.
As students turned their creative talents on these topics we decided to open this repository of software devoted to understanding them. Our hope is that it will develop into a useful resource for students and faculty in both disciplines. Please contribute your good material!
The following is a list of mathematical topics used in computer graphics. The original list was provided by Drexel professor David Breen.
 2D, 3D, 4D real spaces; affine subspaces; homogeneous coordinates
 Vectors and Matrices
 Graphics primitives
 2D primitives developed from triangles: fans, strips
 Convexity and convex sums
 3D models based on 2D faces
 Do we save machine representations for the CG course?
 Transformations

 Primitive geometric transformations
 Creating general transformations via sequences of primitives
 Inverse transformations via primitives
 Do we include transformation stacks?
 This might well be phrased in terms of the viewing transformation
 Hierarchical coordinate systems
 Geometry
 Implicit geometry  lines, circles, ellipses
 Implicit Surfaces  quadrics, superquadrics
 Implicit Equations
 Parametric geometry  lines, circles, ellipses
 Curves (CatmullRom, Bezier, Bspline)
 Parametric surfaces  quadrics, superquadrics, others
 Bezier patches
 Subdivision surfaces
 Procedural Image
 Techniques of computing them from analytic and nonanalytic cases
 Silhouette Edges
 Procedural texture maps
 Noise
 Rayobject intersection
 Bounding Volumes  Bounding spheres and boxes
 Perspective and parallel planar projections
 Nonplanar projections
 Edge detection
 Fourier analysis and convolution
More examples may be found in the lecture slides of CS 430.