Modeling 101

1. Modeling 101 • For the moment assume that all geometry consists of points, lines and faces • Line: A segment between two endpoints • Face: A planar area bounded by line segments • Any face can be triangulated (broken into triangles)

2. Modeling and OpenGL • In OpenGL, all geometry is specified by stating which type of object and then giving the vertices that define it • glBegin(…) …glEnd() • glVertex[34][fdv] • Three or four components (regular or homogeneous) • Float, double or vector (eg float[3]) • Chapter 2 of the red book

3. Rendering • Determine where each object should go in the image • Determine which object is in front at each pixel • Determine what color it is

4. Graphics Pipeline Watt Ch 5 and 6 • Graphics hardware employs a sequence of coordinate systems • The movement of geometry through these spaces is considered a pipeline Local Coordinate Space World Coordinate Space View Space 3D Screen Space Display Space

5. Local Coordinate Space • It is easiest to define individual objects in a local coordinate system • For instance, a cube is easiest to define with faces parallel to the coordinate axis • Key idea: Object instantiation • Define an object in a local coordinate system • Use it multiple times by copying it and transforming it into the global system

6. Global Coordinate System • All the objects in the world are transformed into one coordinate system - the global coordinate system • Lighting is defined in this space • The camera is defined with respect to this space • Some higher level operations, such as advanced visibility computations, can be done here

7. View Space • Associate a set of axes with the image plane • One normal to the image plane • One up in the image plane • One right in the image plane • Some camera parameters are easiest to define in this space • Focal length, image size

8. 3D Screen Space • Transform view space into a cube • Parallel sides make things easier • Task to do: • Rasterization - decide which pixels are covered • Hidden surface removal - decide what is in front • Shading - decide what color things are

9. Display Space • Convert the virtual screen into real screen coordinates • Drop the depth coordinates and translate • The windowing system takes care of this

10. OpenGL and Transformations • OpenGL combines all the transformations up to view space into the MODELVIEW matrix • View space to Screen Space is done with the PROJECTION matrix • Matrix calls multiple some matrix M onto the current matrix C: CM • Set view transformation first, then set transformations from local to world space

11. Defining View Space • View space is defined by location of three mutually perpendicular axes in world space • Translation, rotation and scaling can take points in world space to points in view space • Typically defined by: • Center of the image plane in world space: View Reference Point (VRP) • The normal to the image plane: View Plane Normal (VPN) • A vector in world space that should be “up” in view space (VUP)

12. z View reference point and view plane normal specify film plane. Up vector gives an “up” direction in the film plane. vector v is projection of up vector into film plane = (n x vup) x n u is chosen so that (u, v, n) is a right handed coordinate system; i.e. it is possible to rotate (x->u, y->v, z->n) (and we’ll do this shortly). VRP, VPN, VUP must be in world coords W o r l d c o o r d i n a t e s x y U p v e c t o r V i e w r e f e r e n c e u p o i n t v n V i e w p l a n e n o r m a l

13. World to View Space • Translate by subtracting VRP • Rotate by amount that aligns camera axes with world axes: • All done for you in OpenGL: • gluLookAt takes the VRP, a point along the VPN, and VUP • Multiplies the required transformation onto the current transformation (normally the identity)

14. Default OpenGL Camera • The default OpenGL image plane has u aligned with the x axis, v aligned with y, and n aligned with z • Means the default camera looks along the negative z axis • Makes it easy to do 2D drawing (no need for any view transformation)

15. Left vs Right Handed View Space • You can define u as right, v as up, and n as toward the viewer: a right handed system uv=n • Advantage: Standard mathematical way of doing things • You can also define u as right, v as up and n as into the scene: a left handed system vu=n • Advantage: Bigger n values mean points are further away • OpenGL is right handed

16. Projection • The conversion from view space to screen space is called projection • Two general classes: • Orthographic, or parallel, projection • Perspective projection

17. Orthographic Projection • Points project along rays perpendicular to the image plane • Just drop the n coordinate, and maybe scale and translate • OpenGL: glOrtho(…) • Sets the appropriate projection matrix

18. Orthographic projection

19. Abstract camera model - box with a small hole in it Pinhole cameras work in practice - camera obscura, etc Perspective Projection

20. Distant Objects Are Smaller

21. Parallel lines meet common to draw film plane in front of the focal point

22. Vanishing points • Each set of parallel lines (=direction) meets at a different point: The vanishing point for this direction • Sets of parallel lines on the same plane lead to collinear vanishing points: the horizon for that plane • Easy examples • corridor • higher = further away • Good way to spot faked images