Part B - Direct3D

Three-Dimensional Mathematics

Extend vector and matrix concepts to 3 dimensions
Build a library to support vector and matrix syntax and operations between them
Describe the transformations for converting from local space into clip space
Describe Direct3D vector and matrix utilities


Coordinate Systems | Vectors | Basic Transformations | Math Library
Frame | General Transformations | Direct3D Utilities |Exercises



Introducing a third dimension into the model of a scene enhances its realism, admits motion of the viewer relative to the scene, and allows for perspective on the objects in the scene.  In a two-dimensional model, sprites move along a flat surface, the viewer is outside the scene, and there is no perspective.  The representations of sprite positions, orientations, rotations, scalings, and translations across the surface are straightforward.  But once we add a third dimension, the mathematics becomes more complex.  There are two new degrees of freedom: the objects in the scene can move around and in front of one another and the observer can move towards, away from, and around the objects in the scene. 

In this chapter, we extend the concepts defined in the previous chapter to three dimensional coordinate systems.  We introduce the two system for describing motion, the viewpoint transformation, and the perspective transformation.  We introduce the framework's vector and matrix library, which defines the structures and operations on them.  We conclude by describing the Direct3D structures and utilities available as alternatives to that library. 


Coordinate Systems

Two rectangular coordinate systems for 3-dimensional representations follow directly from our 2-dimensional coordinate system.  These systems differ from one another by the direction of the added axis, which identifies the system's handedness as illustrated in the figure below:

  • a left-handed system - the added axis is directed into the page away from the observer
  • a right-handed system - the added axis is directed out of the page towards the observer

The system axes are labelled x, y, and z.  They are perpendicular to one another (mathematicians use the term orthogonal) and share a common point - the origin.  The triad itself can be oriented in any way.  The convention in 3-d game programming is to keep the x-y plane parallel to the screen with the x axis directed horizontally towards the right and the y axes vertically upwards. 

handedness

Under this convention, the x and y axes define a plane that is identical in both systems (as well as in two-dimensional representations). 

The thumb and forefinger of the hand associated with a system determine the direction of its z axis.  Aligning the thumb with the positive x-axis and the forefinger with the positive y-axis, the remaining fingers point in the direction of the positive z-axis axis. 

The handedness of a system is a property of the system.  No combination of rotation, scaling and translation transformations can convert handedness.  Engineering and scientific applications typically use the right-handed system, which has been the most written about given its predominance over the last 60 years.  The left-handed system is more convenient for gaming applications. 

The difference between the two systems appears in the symbolic descriptions of vector transformations.  The vector-matrix-vector form of multiplication is compatible with the right-handed system.  The vector-vector-matrix form of multiplication is compatible with the left-handed system. 

We focus on the left-handed system in this chapter and use it throughout all of the remaining samples. 


Vectors

A vector in a three-dimensional coordinate system is completely defined by three component values.  Each value specifies a distance along an axis of the system and describes the difference between the vector's head and tail positions in the positive direction of that axis: 

 pT = [px py pz]T
    = [hx-tx hy-ty hz-tz]T

The vector's magnitude or length follows from Pythagoras' theorem:

 ||p|| = √[(px)2 + (py)2 + (pz)2]

The components of a vector's reverse are the negative values of the original components:

 rT = -pT = [-px -py -pz]T

Multiplying a vector by a scalar applies the scalar multiplier to each component: 

 rT = spT = [spx spy spz]T

Adding two vectors produces a third vector with components equal to the sum of the components of the contributing vectors: 

 rT = pT + qT
    = [px py pz]T + [qx qy qz]T
    = [px+qx py+qy pz+qz]T

If, in an addition, the tail of the added vector coincides with the head of the reference, vector, the result is a vector from the tail of the reference to the head of the added vector:

 rT = pT + qT
    = [hpx-tpx hpy-tpy hpz-tpz]T + [hqx-hpx hqy-hpy hqz-hpz]T
    = [hpx-tpx+hqx-hpx hpy-tpy+hqy-hpy hpz-tpz+hqz-hpz]T
    = [hqx-tpx hqy-tpy hqz-tpz]T

Subtracting one vector from another vector produces a third vector with components equal to the difference of the components of the two vectors: 

 rT = pT - qT
    = [px py pz]T - [qx qy qz]T
    = [px-qx py-qy pz-qz]T

If, in a subtraction, the tails of the two vectors coincide, the result is a vector from the head of the subtracted vector to the head of the reference vector:

 rT = pT - qT
    = [hpx-tpx hpy-tpy hpz-tpz]T - [hqx-tpx hqy-tpy hqz-tpz]T
    = [hpx-tpx-(hqx-tpx) hpy-tpy-(hqy-tpy) hpz-tpz-(hqz-pz)]T
    = [hpx-hqx) hpy-hqy hpz-hqz]T

The scalar or dot product of the two vectors is given by

 s = p∙q
   = pTq
                |qx|
   = [px py pz] |qy|
                |qz|

   = pxqx + pyqy + pzqz

The product of a row vector and a column vector is the sum of the products of a column coefficient of the row vector with the corresponding row coefficient of the column vector:

 [a b c] | d | = ad + be + cf
         | e |
         | f |

The dot product of a vector with itself is the square of its magnitude:

 s = p∙p
   = pTp
                | px |
   = [px py pz] | py |
                | pz |

   = (px)2 + (py)2 + (pz)2
   = ||p||2

The norm of the vector is a vector of unit magnitude with the same direction:

 n = p / ||p||
   = p / (p ∙ p)1/2

The vector or cross product of two 3-dimensional vectors is given by:

 rT = pT × qT
    = [pyqz - pzqy  pzqx - pxqz  pxqy - pyqx]T

The direction of the cross product in a left-handed system is given by the left-hand rule.  If the thumb on the left hand points in the direction of the left operand and the forefinger points in the direction of the right operand, then the direction of the cross product is the direction of the remaining fingers.  (For a right-handed system, we use the right hand for the right-hand rule.) 


Basic Transformations

For compatibility across transformations - rotations, scalings, and translations - matrices in three-dimensional space are 4 x 4 matrices.  We expand our component description of a vector to include a fourth unit value:

 pT = [px py pz 1]T

A matrix describing any transformation in three-dimensional space has the general form:

 | m11 m12 m13 m14 |
 | m21 m22 m23 m24 |
 | m31 m32 m33 m34 |
 | m41 m42 m43 m44 |

The first index refers to the row number and the second index to the column number. 

To preserve the unit component of the expanded vector in any vector-vector-matrix multiplication, the elements of the fourth column are

 | m11 m12 m13 0 |
 | m21 m22 m23 0 |
 | m31 m32 m33 0 |
 | m41 m42 m43 1 |

Rotations

The direction of positive rotation depends upon the handedness of the adopted system.  The left-hand rule identifies the positive direction of rotation for left-handed systems.  Grasping the positive side of the axis of rotation with the thumb directed away from the origin, the direction from the palm towards the fingertips is the positive angular direction.

Rotation about the z axis

The transpose of the matrix for rotation about the z axis (RT) takes the form:

 |  cos θ  sin θ     0     0 |
 | -sin θ  cos θ     0     0 |
 |   0      0        1     0 |
 |   0      0        0     1 |

θ is positive-valued measured counter-clockwise about the z axis when facing in the positive direction of that axis. 

Rotation about the x axis

The transpose of the matrix for rotation about the x axis (RT) takes the form:

 | 1      0        0       0 |
 | 0     cos θ    sin θ    0 |
 | 0    -sin θ    cos θ    0 |
 | 0      0        0       1 |

θ is positive-valued counter-clockwise about the x axis when facing in the positive direction of that axis. 

Rotation about y axis

The transpose of the matrix for rotation about the y axis (RT) takes the form:

 | cos θ    0    -sin θ    0 |
 |  0       1      0       0 |
 | sin θ    0     cos θ    0 |
 |  0       0      0       1 |

θ is positive-valued counter-clockwise about the y axis when facing in the positive direction of that axis. 

Scaling

The transformation matrix for three-dimensional scaling takes the form:

 | m11 0   0   0 |
 | 0   m22 0   0 |
 | 0   0   m33 0 |
 | 0   0   0   1 |

Translation

The transpose of the translation matrix (TT) takes the form:

 |  1   0   0  0 |
 |  0   1   0  0 |
 |  0   0   1  0 |
 | ∆x  ∆y  ∆z  1 |

where ∆x, ∆y, and ∆z denote translations in positive directions along the x, y, and z axes respectively. 


Math Library

The framework has its own library to support vectors, matrices, and operations on them.  The MathDecl.h header file defines Vector and Matrix structures.  The MathDef.h header file defines the member operators, helper functions and helper operators.  All of the definitions are inline. 

Vector

 struct Vector {
     float x;
     float y;
     float z;
     Vector() : x(0), y(0), z(0) {}
     Vector(float xx, float yy, float zz) : x(xx), y(yy), z(zz) {}
     Vector  operator+();
     Vector  operator-();
     Vector& operator+=(const Vector& a);
     Vector& operator-=(const Vector& a);
     Vector  operator*(const Matrix& m);
     Vector  operator*(const Matrix& m) const;
     Vector  operator*=(const Matrix& m);
     float   length() const;
 };

Note that the library omits the fourth, unit-value component.  It does nevertheless accounts for it in the matrix multiplications.

Matrix

 struct Matrix {
     float m11, m12, m13, m14;
     float m21, m22, m23, m24;
     float m31, m32, m33, m34;
     float m41, m42, m43, m44;
     Matrix() : m11(0), m12(0), m13(0), m14(0),
                m21(0), m22(0), m23(0), m24(0),
                m31(0), m32(0), m33(0), m34(0),
                m41(0), m42(0), m43(0), m44(0) {}
     Matrix(float x) : m11(x), m12(0), m13(0), m14(0),
                       m21(0), m22(x), m23(0), m24(0),
                       m31(0), m32(0), m33(x), m34(0),
                       m41(0), m42(0), m43(0), m44(1) {}
     Matrix(const Vector& v) : m11(v.x), m12(0), m13(0), m14(0),
                               m21(0), m22(v.y), m23(0), m24(0),
                               m31(0), m32(0), m33(v.z), m34(0),
                               m41(0), m42(0), m43(0), m44(1) {}
     Matrix(float i11, float i12, float i13, float i14,
            float i21, float i22, float i23, float i24,
            float i31, float i32, float i33, float i34,
            float i41, float i42, float i43, float i44) :
                m11(i11), m12(i12), m13(i13), m14(i14),
                m21(i21), m22(i22), m23(i23), m24(i24),
                m31(i31), m32(i32), m33(i33), m34(i34),
                m41(i41), m42(i42), m43(i43), m44(i44) {}
     Matrix& isIdentity();
     Matrix& operator+=(const Matrix& a);
     Matrix& operator-=(const Matrix& a);
     Matrix& operator*=(const Matrix& m);
     Matrix  transpose() const;
     Matrix& translate(float x, float y, float z);
     Matrix& rotatex(float rad);
     Matrix& rotatey(float rad);
     Matrix& rotatez(float rad);
     Matrix& rotate(const Matrix& rot);
     Matrix& scale(float sx, float sy, float sz);
     Matrix& orient(const Matrix& rot);
     Vector  position() const;
     Matrix  rotation() const;
     Vector  direction(const Vector& v) const;
     Vector  direction(char c) const;
     Matrix& operator*=(const Matrix& m);
     Matrix  transpose() const;
 };

Vector Matrix Multiplication

The vector-matrix multiplication uses vector-vector-matrix form for compatibility with a left-handed coordinate system.  A Vector reads as a row vector and a Matrix reads as the transpose of the matrix written to the engineering and scientific convention.  For example, for the following transformation of a vector by a matrix:

 qT = pT RT

we would write

 Vector p, q;
 Matrix R;

 q = p * R;  // transpose symbols omitted

For brevity, the code throughout the framework omits the transpose symbols.

Rotation

The rotation of an object is typically a rotation about the origin of its local frame of reference and not about the origin of the world frame of reference.  If the origin of the object's local frame of reference does not coincide with the origin of the world frame of reference, a rotation about the local frame of reference will differ from the same rotation about world space. 

To achieve a rotation about the origin of the local frame of reference, the library functions perform a three-part operation. 

  • translate the object to the origin of the world frame of reference. 
  • rotate the object about the origin of the world/local frame. 
  • translate the object back to its position prior to the rotation. 

This is simply done by extracting the coordinates of the fourth row of the object's world matrix, rotating, and inserting them into the result. 

For example, the rotatez() method on a Matrix object applies a rotation about the z axis of rad radians to the current matrix and returns a reference to the rotated matrix:

 inline Matrix& Matrix::rotatez(float rad) {
    float c = cosf(rad);
    float s = sinf(rad);
    Matrix rot(c, s, 0, 0,
              -s, c, 0, 0,
               0, 0, 1, 0,
               0, 0, 0, 1);
    rotate(rot);
    return *this;
 }
 inline Matrix& Matrix::rotate(const Matrix &rot) {
     float x = m41, y = m42, z = m43;
     m41 = m42 = m43 = 0; // translate to world origin
     *this *= rot;
     m41 = x;             // translate back to original position
     m42 = y;
     m43 = z;
     return *this;
 }

Scaling

The scaling of an object is typically a scaling about the origin of its local frame of reference and not about the origin of the world frame of reference. 

To achieve a pure scaling about the origin of the local frame of reference, the library performs a three-part operation. 

  • translate the object to the origin of the world frame of reference. 
  • scale the object about the origin of the world/local frame. 
  • translate the object back to its position prior to the scaling. 

As with rotations, this is done by extracting the coordinates of the fourth row of the object's world matrix, scaling, and inserting them into the result. 


Frame

The framework uses the Frame class to store the transformation matrix for any design element.  Classes for all design elements that have positional or orientational properties derive from this base class.

A Frame object holds the matrix for transforming from the design element's local frame of reference to its parent's frame of reference.  The parent frame of reference can be the child frame of another parent.  If no parent frame of reference is specified, the frame of reference is world space. 

The Frame class exposes thirteen virtual methods:

  • translate() - translates the Frame object
  • scale() - scales the Frame object
  • rotatex() - rotates the Frame object about an axis parallel to the x world axis
  • rotatey() - rotates the Frame object about an axis parallel to the y world axis
  • rotatez() - rotates the Frame object about an axis parallel to the z world axis
  • rotate() - rotates the Frame object about a Vector
  • position() - returns the world position of the Frame object
  • rotation() - returns the world orientation of the Frame object
  • orientation() - returns a Vector with the direction of a local axis of the Frame object
  • world() - returns the world transformation for the Frame object
  • attachTo() - attaches the current object to the parent pointed to by Frame*

The Frame class holds the address of the parent object, if any, and the transformation matrix with respect to the parent frame of reference (or world space):

 // Frame.h

 class Frame {
     Matrix T;      // transformation to parent frame
     Frame* parent; // points to the parent frame, if any
   public:
     Frame();
     virtual void scale(float sx, float sy, float sz)   { T.scale(sx, sy, sz); }
     virtual void rotatex(float rad)                    { T.rotatex(rad); }
     virtual void rotatey(float rad)                    { T.rotatey(rad); }
     virtual void rotatez(float rad)                    { T.rotatez(rad); }
     virtual void rotate(const Vector& axis, float rad) { T.rotate(axis, rad); }
     virtual void translate(float x, float y, float z)  { T.translate(x, y, z); }
     virtual void orient(const Matrix& rot)             { T.orient(rot); }
     virtual Vector position() const;
     virtual Matrix rotation() const;
     virtual Vector orientation(const Vector& v) const;
     virtual Vector orientation(char c) const;
     virtual Matrix world() const;
     virtual void   attachTo(Frame* newParent);
     virtual ~Frame() {}
 };

The inline methods call the equivalent methods on the Matrix object.

Implementation

Construct

The Frame constructor initializes the transformation matrix to the identity matrix with no parent:

 Frame::Frame() : T(1), parent(0) {}

orientation

The orientation() method takes a char returns the normalized free vector that describes the direction of the specified local axis:

 Vector Frame::orientation(char axis) const {
     Vector v;
     switch(axis) {
         case 'x': v = Vector(1, 0, 0); break;
         case 'y': v = Vector(0, 1, 0); break;
         case 'z': v = Vector(0, 0, 1); break;
     }
     return orientation(v);
 }

The orientation() method that takes a Vector reference returns the coordinates of that Vector in world space:

 Vector Frame::orientation(const Vector& v) const {
     return v * rotation();
 }

position

The position() method returns the position of the local frame of reference in world space:

 Vector Frame::position() const {
     return parent ? ::position(T) * parent->rotation() + parent->position() :
      ::position(T);
 }

rotation

The rotation() method returns the orientation of the local frame of reference in world space:

 Matrix Frame::rotation() const {
     return parent ? ::rotation(T) * parent->rotation() : ::rotation(T);
 }

world

The world() method returns the world transformation matrix:

 Matrix Frame::world() const {
     return parent ? T * parent->world() : T;
 }

attachTo

The attachTo() method returns a Matrix that holds the world transformation matrix:

 void Frame::attachTo(Frame* newParent) {
     if (parent) T = world(); // detach from current parent, if any
     parent = newParent;      // attach to newParent
     if (parent) {
         Matrix m = parent->rotation(); // extract orientation
         m = m.transpose();
         T.rotate(m);                   // invert orientation
         // express offset in local coordinates wrt to parent frame
         Vector p = (::position(T) - parent->position()) * m;
         T.m41 = p.x;
         T.m42 = p.y;
         T.m43 = p.z;
     }
 }

Note that the inverse of a rotation matrix is the same as its transpose.


General Transformations

The position and orientation of each design element with respect to the screen coordinates depends upon the element's position and orientation in world space, the viewer's position and orientation, the perspective on the scene, and the the reference frame of the screen.  The game design itself describes the position and orientation for each design element in world space.  Four transformations are involved in determining the position in screen space of any point on a design element:

  • world transformation - transforms local coordinates to world coordinates
  • view transformation - transfroms world coordinates to camera space coordinates
  • projection transformation - transfroms camera coordinates to clip space coordinates
  • viewport transformation - transfroms clip space coordinates to screen coordinates

World Transformation

The graphic repesentation of a design element consists of points identified by vectors bound to its centroid.  The origin of the local coordinate system is at the element's centroid.  We describe each point on the element in terms of the bound vector to that point in the local coordinate system.  We assume that the element is rigid, so that its bound vectors do not change throughout the element's lifetime. 

Each design element has its own homogeneous matrix.  This matrix describes the element's position, orientation, and scaling with respect to its parent; in the absence of any parent, with respect to world space.  To calculate the world position of a point on the element, we multiply the bound vector with the element's world transformation matrix.  The graphics hardware performs this transformation. 

As a design element translates or rotates, its world matrix changes.  Its current world matrix is a concatenation of successive rotation, scaling, and translation matrices (along with those of its parents): 

 WTi+1 = WTi QTi

where Qi is the current transformation (rotation, scaling, translation) matrix and i is an index that identifies the transformation in the sequence of successive world transformations (i = 0, ..., i) from the original position and orientation.

Camera Transformations

The view transformation defines the position and orientation of the viewpoint in world space.  An identity transformation describes a viewpoint at the origin of world space with its local axis aligned with the world axes, and the look at point along the positive world z axis.  We use three vectors to identify the current viewpoint and its orientation in world space:

  • position
  • look at direction
  • up direction

We convert these three vectors into the homogeneous view matrix (V).  The graphics hardware uses this matrix to transform the world coordinates of each point on each graphic representation into a camera-space coordinates. 

Given the position (p), the look at point (d) and the up (u) direction in world space, three normal vectors to determine the view matrix:

 z =   (d - p) / ||(d - p)||     // left-handed system
 x = u × z / ||u × z||
 y = z × x

projection onto clipping space

The transpose of the view matrix is

 |    xx    yx    zx  0 |
 |    xy    yy    zy  0 |
 |    xz    yz    zz  0 |
 |  -r∙p  -y∙p  -z∙p  1 |

The source code for this transpose is:

 inline Matrix view(const Vector& p, const Vector& d, const Vector& u) {
     Vector z = normal(d - p);
     Vector x = normal(cross(u, z));
     Vector y = cross(z, x);
     return Matrix(       x.x,        y.x,        z.x,  0,
                          x.y,        y.y,        z.y,  0,
                          x.z,        y.z,        z.z,  0,
                   -dot(x, p), -dot(y, p), -dot(z, p),  1);
 }

Projection Transformation

The projection transformation introduces perspective and scaling on the different objects in a scene.  Without perspective objects that are further away appear larger than they should.  The projection transformation makes objects that are further away appear smaller than they are and reduces objects that are sufficiently far away to single points or invisible ones.  The projection transformation excludes objects that are too close or outside the viewing angle.  The result of a projection transformation (P) is a conversion from a camera space view to clip space view.  Clip space is a right cuboid with its near face centered on the origin and its far face centered on [0,0,1] and the faces normal to the z axis square of side length 2 as shown below. 

projection onto clipping space

The projection matrix transforms any point in camera space to a point in clip space.  The transformation involves a scaling and a perspective projection.  We call the prism that contains the points that map into the right cuboid the frustum.  Points outside this frustum are outside clip space and will not appear on the screen.  We define the persective in terms of a field of view angle at the camera.  This angle defines the inclination of the frustum's sides.

Four values completely define the projection matrix:

  • the z coordinate in camera space of the near-clipping plane - the cutoff plane for points too close to the viewpoint
  • the z coordinate in camera space of the far-clipping plane - the cutoff plane for points too far from the viewpoint
  • the field of view angle
  • the aspect ratio of the display

frustum for the projection matrix

The source code for the transpose of the projection matrix is

 inline Matrix projection(float fov, float aspect, float near_cp,
  float far_cp) {
     float sy = 1.0f / tanf(fov * 0.5f);
     float sx = sy / aspect;
     float sz = far_cp / (far_cp-near_cp);
     return Matrix(sx,  0,             0, 0,
                    0, sy,             0, 0,
                    0,  0,            sz, 1,
                    0,  0, -near_cp * sz, 0);
 }

Viewport Transformation

The viewport transformation converts clip space coordinates into screen coordinates.  Screen coordinates are defined in two dimensions with the origin in the top left corner and the x and y axis to the right and down respectively.  Six factors completely determine the coefficients of this matrix (Y):

  • w - width of the screen in pixels
  • h - height of the screen in pixels
  • x - pixel x coordinate of the top left corner (0)
  • y - pixel y coordinate of the top left corner (0)
  • minz - minimum of the range of depth values (0.0)
  • maxz - maximum of the range of depth values (1.0)

The transpose of the viewport matrix is

 |   w/2     0         0       0 |
 |    0    -h/2        0       0 |
 |    0      0    maxz - minz  0 |
 | x + w/2 y + h/2   minz      1 |

The graphics hardware converts clip space coordinates to screen coordinates using this transformation. 

Composite Transformation

The composite transformation matrix for each point on each object is the product of the world, view, projection, and viewport matrices:

 (Xcomposite)T = WT VT PT YT

This matrix converts vectors that define points on drawable objects in local coordinate system into points on screen space.


Direct3D Utilities

The Direct3D utility library provides declarations and definitions for vectors and matrices, including the view and projection matrices for DirectX applications.  The Direct3D syntax cannot be used with the OpenGL API.  The Direct3D identifiers are also upper-case, which makes coding all the more difficult to read.  The declarations and definitions are part of the D3DX utility library.

The D3DX library functions operate on D3DX structures.  The structure types are:

  • D3DXVECTOR3 - 3 component vector (.x, .y, .z)
  • D3DXMATRIX - 4 x 4 homogeneous matrix (._11, ._12, ._13, ._14, ._21, ... )

The D3DX functions corresponding to the operations implemented in the framework's math library are listed below. 

Operation MathDefinitions.h Function D3DX Function
Vector Operators and Functions
reverse Vector Vector::operator-() const D3DXVECTOR3 D3DXVECTOR3::operator-() const
add Vector& Vector::operator+=(const Vector&) D3DXVECTOR3& D3DXVECTOR3::operator+=(const D3DXVECTOR3&)
subtract Vector& Vector::operator-=(const Vector&) D3DXVECTOR3& D3DXVECTOR3::operator-=(const D3DXVECTOR3&)
magnitude float Vector::length() const float D3DXVec3Length(const D3DXVECTOR3*)
add Vector operator+(const Vector&, const Vector&) D3DXVECTOR3 D3DXVECTOR3::operator+(const D3DXVECTOR3&) const
subtract Vector operator-(const Vector&, const Vector&) D3DXVECTOR3 D3DXVECTOR3::operator-(const D3DXVECTOR3&) const
multiply Vector operator*(float, const Vector&) D3DXVECTOR3 D3DXVECTOR3::operator*(float) const
divide Vector operator/(const Vector&, float) D3DXVECTOR3 D3DXVECTOR3::operator/(float) const
scalar product float dot(const Vector&, const Vector&) float D3DXVec3Dot(const D3DXVECTOR3*, const D3DXVECTOR3*)
normal Vector normal(const Vector&) D3DXVECTOR3* D3DXVec3Normalize(D3DXVECTOR3*, const D3DXVECTOR3*)
vector product Vector cross(const Vector&, const Vector&) D3DXVECTOR3* D3DXVec3Cross(D3DXVECTOR3*, const D3DXVECTOR3*, const D3DXVECTOR3*)
Matrix Operators and Functions
transform Vector& Vector::operator*=(const Matrix&) D3DXVECTOR3* D3DXVec3TransformCoord(D3DXVECTOR3*, const D3DXVECTOR3*, const D3DXMATRIX*)
initialize Matrix& Matrix::isIdentity() D3DXMATRIX* D3DXMatrixIdentity(D3DXMATRIX*)
multiply Matrix& Matrix::operator*=(const Matrix&) D3DXMATRIX* D3DXMATRIX::operator*=(const D3DXMATRIX&)
transpose Matrix Matrix::transpose() D3DXMATRIX* D3DXMatrixTranspose(D3DXMATRIX*, const D3DXMATRIX*)
multiply Vector operator*(const Vector&, const Matrix&) D3DXVECTOR3* D3DXVec3TransformCoord(D3DXVECTOR3*, const D3DXVECTOR3*, const D3DXMATRIX*)
translate Matrix Matrix::translate(float, float, float) D3DXMATRIX* D3DXMatrixTranslation(D3DXMATRIX*, float, float, float)
rotate about x axis Matrix Matrix::rotatex(float) D3DXMATRIX* D3DXMatrixRotationX(D3DXMATRIX*, float)
rotate about y axis Matrix Matrix::rotatey(float) D3DXMATRIX* D3DXMatrixRotationY(D3DXMATRIX*, float)
rotate about z axis Matrix Matrix::rotatez(float) D3DXMATRIX* D3DXMatrixRotationZ(D3DXMATRIX*, float)
scale Matrix Matrix::scale(float, float, float) D3DXMATRIX* D3DXMatrixScaling(D3DXMATRIX*, float, float, float)

The D3DXMatrixLookAtLH() function returns the transpose of the view matrix for a left-handed system:

     D3DXMATRIX view;
     D3DXVECTOR3 at = viewpt + direction;
     D3DXVECTOR3 up = D3DXVECTOR3(0, 1, 0);
     D3DXMatrixLookAtLH(&view, &viewpt, &at, &up);

The D3DXMatrixPerspectiveFovLH() function returns the transpose of the projection matrix for a left-handed system:

     D3DXMATRIX projection;
     D3DXMatrixPerspectiveFovLH(&projection, FIELD_OF_VIEW,
      width/float(height), NEAR_CLIPPING, FAR_CLIPPING);

Exercises

  • Read the DirectX Documentation on Transforms
  • Read the Wikipedia article on Transformations



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