Direct3D Workshop November 17, 2005 Workshop by Geoff Cagle Presented by Players 2 Professionals

Workshop Breakdown ► Part 1: 3D Graphics Pipeline ► Part 2: Projective Space ► Part 3: Windows Programming ► Intermission ► Part 4: Setting Up Direct3D ► Part 5: Using Direct3D

Notes and Code Samples ► You can download these notes and all the code samples from

Part 2 Projective Space

Motivation ► How should space be represented in computer graphics? ► Using linear algebra seems like a fairly natural way to represent space, but eventually you will find it has its limits. ► A better solution is called projective space, and it’s the math being used behind all graphics cards today. ► Let’s start with the linear algebra and develop projective spaces from there.

Vectors ► Remember vectors? ► They have direction and magnitude. ► We usually break down vectors into x, y, and z components. ► Vectors can represent coordinates (displacement from an origin).

Matrices ► Remember matrices? ► Matrices can represent transformations  From a vector to a vector.  From another matrix to another matrix. ► Matrices can also represent tensors, orientations, or coordinate systems.

Matrix-Vector Multiplication ► Each columns of the matrix can be interpreted as transforming a component of the vector.

Linear Transformations ► A transformation T is linear iff ► Linear transformations include Scaling, Shearing, and Rotations. ► Translation is not linear:

Linear Transformations (Cont’d) ► A transformation is linear iff it can be written in matrix- vector form:  You can find the columns of A by transforming the standard basis: ► Multiplying matrices together results in a composition of transforms.

Affine Spaces ► Extending vector spaces, affine spaces add another set of objects called points. ► A point has a position, but not a direction or magnitude. ► Points have only a couple operations:  P – Q = v  Q + v = P  Where P and Q are points and v is a vector. ► You cannot add two points together or multiply a point by a scaler.

Affine Frame ► Let e 1, e 2, e 3 form a basis for the vectors in an affine space, A. ► Let O be a point in A. We will call this point the origin. ► Then e 1, e 2, e 3, O forms a frame for A.  Any vector in A can be written as:  Any point can be written as:

Affine Coordinates ► Using frames we can write vectors like this: ► And points like this:

Affine Coordinates ► The appended 0 or 1 is called the w-component. ► In affine space this is either  0 for a vector  1 for a point  And anything else is undefined. ► Note that if you add vectors together or scale them, the result is a vector, but if you try the same with points, the result is undefined.

Affine Transform ► An affine transform adds translation to a linear transform. ► In regular vector space it has the form: ► In affine space it is written as: ► If you run a point through this transform it is translated, while vectors are not.

Perspective ► Projective space is an extension to affine space. ► Using transformations in this space we will be able to calculate perspective.

Projective Space ► In projective space, the w-component is either 0 for a vector or any nonzero real number for a point. ► However, coordinates in projective space are homogeneous.  Two points in projection space are said to be equivalent if the coordinates of one are a multiple of the other.  That is, if:

Projective Space (Cont’d) ► Points in projective space actually represent lines through the origin. ► We can standardize the points, by dividing by w so that its w component is 1.

Projective Space (Cont’d) ► In the case of the projective plane (the 2D projective space).  3D coordinates are used.  The intersection points of the lines through the origin and the plane w=1 make a perspective image of the space.

Projective Transform ► The projective transform generalizes the affine transform. ► With homogeneous coordinates it looks like: ► With linear algebra it would look like this:

Perspective Transform ► When perspective is actually calculated in computer graphics, it doesn’t usually flatten the space. ► Instead coordinates are transformed so that the viewing frustum is transformed into a unit square ► This is done through a composite of a scaling and a projective transform.

Direct3D Matrices and Vectors ► Direct3D includes a matrices and vectors (as well as planes and quaternions).  D3DVECTOR and D3DMATRIX  D3DXVECTOR # and D3DMATRIX #  D3DXMATRIXA #  Where # is an integer. ► List of functions for D3DX* structures =/library/en- us/directx9_c/directx/graphics/reference/d3dx/fun ctions/math/mathfunctions.asp =/library/en- us/directx9_c/directx/graphics/reference/d3dx/fun ctions/math/mathfunctions.asp =/library/en- us/directx9_c/directx/graphics/reference/d3dx/fun ctions/math/mathfunctions.asp

Direct3D Math Quirks ► Be aware of Direct3D’s left handed coordinate system  X is positive to the left  Y is positive to the top  Z is position going into the monitor ► Matrices are multiplied on the right side of a vector:

References 1. Computer-Aided Geometric Design by F. Yamaguchi 2. Computer Animation Slides by JeHee Lee Computer Graphics Notes by Ken Joy /GraphicsNotes/Graphics-Notes.html /GraphicsNotes/Graphics-Notes.html /GraphicsNotes/Graphics-Notes.html