1. Honours Graphics 2008 Session 3

2. Today’s focus • Perspective and orthogonal projection • Quaternions • Graphics camera

3. Projections • Visualizing 3D data on a 2D screen requires projection of the data onto a 2D plane • Several projection models exist, with varying qualities • We’re interested in orthogonal and perspective projection

4. Perspective projection

5. Perspective projection, cont. • Starting from world coordinate space:

6. Increasing focal length and distance of the camera to infinity changes perspective into orthogonal projection

7. Orthogonal projection • Flat, 2D projection • Used when precise profiles or measurements need to be displayed • Special case of perspective projection, when focal length approaches infinity

8. Quaternions • Developed by Sir William Rowan Hamilton in 1843 • Generally superceded by vectors and matrices, but still very useful in applied mathematics and computer graphics • Specifically useful to compute 3D rotations • Consists of scalar and “vector” components

9. Quaternions, cont. • Defined as an extension to the complex numbers: three components i, j and k all are squareroots of -1, hence • Furthermore

10. Quaternions, cont. • Finally • Quaternion addition and subtraction

11. Quaternions, cont. • Magnitude of a quaternion • Quaternion multiplication

12. Quaternions, cont. • For graphics purposes a unit quaternion is used, which has the property that • Unit quaternions represent rotation / orientation

13. Quaternions, cont. • Quaternion to rotation matrix conversion

14. Graphics camera • Both DirectX and OpenGL expose camera parameters that consist of • View position • View target • Relative view-up direction

15. Camera commands • OpenGLvoid gluLookAt(eyeX, eyeY, eyeZ, centerX, centerY, centerZ, upX, upY, upZ ) • DirectXD3DXMatrixLookAtLH(mat, eye, center, up);DX9Device.SetTransform(D3DTS_VIEW, mat);

16. Homework • Write a camera library that makes use of quaternions. Allow for keyboard and mouse input to control the camera. • For next session