14_line_clip_rasterization

– clip p to plane
• If H•p > 0 and H•q > 0
– pass through
• If H•p < 0 and H•q < 0
– clipped out
n q
MIT EECS 6.837, Durand and Cutler
Clipping against the frustum
H = (A,B,C,D)
MIT EECS 6.837, Durand and Cutler
Clipping a Point with respect to a Plane
• If d = H•p 0 Pass through
• If d = H•p < 0: Clip (or cull or reject)
• Homogenous point: (x,y,z,w)
infinite number of equivalent homogenous coordinates: (sx, sy, sz, sw)
• Homogenous Plane Equation: Ax+By+Cz+D = 0 → H = (A,B,C,D)
far
• User may define additional clipping planes
near
right bottom
MIT EECS 6.837, Durand and Cutler
Why clip?
• Avoid degeneracies
– Don’t draw stuff behind the eye – Avoid division by 0 and overflow
MIT EECS 6.837, Durand and Cutler
Clipping
• Eliminate portions of objects outside the viewing frustum • View Frustum
– boundaries of the image plane projected in 3D left – a near & far clipping plane top
• Efficiency
– Don’t waste time on objects outside the image boundary
• Other graphics applications (often non-convex)
– Hidden-surface removal, Shadows, Picking, Binning, CSG (Boolean) operations (2D & 3D)
The Graphics Pipeline: Line Clipping & Line Rasterization
MIT EECS 6.837, Durand and Cutler
Last Time?
Modeling Transformations Illumination (Shading) Viewing Transformation (Perspective / Orthographic) Clipping Projection (to Screen Space) Scan Conversion (Rasterization) Visibility / Display
MIT EECS 6.837, Durand and Cutler
• Ray Tracing vs. Scan Conversion • Overview of the Graphics Pipeline • Projective Transformations
Questions?
MIT EECS 6.837, Durand and Cutler
• • • •
Overview of Rasterization Line Rasterization Circle Rasterization Antialiased Lines
MIT EECS 6.837, Durand and Cutler
Implicit 3D Plane Equation
q
MIT EECS 6.837, Durand and Cutler
Segment Clipping
• If H•p > 0 and H•q < 0
– clip q to plane
p
• If H•p < 0 and H•q > 0
– clip p to plane
• If H•p > 0 and H•q > 0
Hnear = ( 0 0 –1 –near) Hfar = ( 0 0 1 far ) Hbottom = ( 0 near bottom 0 ) (left, bottom, –near) Htop = ( 0 –near –top 0 ) Hleft = ( left near 0 0 ) Hright = (–right –near 0 0 ) MIT EECS 6.837, Durand and Cutler
• For each frustum plane H
– – – – If H•p > 0 and H•q < 0, clip q to H If H•p < 0 and H•q > 0, clip p to H If H•p > 0 and H•q > 0, pass through If H•p < 0 and H•q < 0, clipped out
MIT EECS 6.837, Durand and Cutler
• Portions of the object outside the view frustum are removed
• Rasterize objects into pixels
Today
• • • • • • Why Clip? Line Clipping Overview of Rasterization Line Rasterization Circle Rasterization Antialiased Lines
H = (A,B,C,D)
Infinite number of equivalent plane expressions: sAx+sBy+sCz+sD = 0 → H = (sA,sB,sC,sD)
MIT EECS 6.837, Durand and Cutler
Point-to-Plane Distance
• Clip (or cull or reject) point p if any H•p < 0
MIT EECS 6.837, Durand and Cutler
What are the View Frustum Planes?
(right*far/near, top*far/near, –far)
MIT EECS 6.837, Durand and Cutler
Clipping strategies
• Don’t clip (and hope for the best) • Clip on-the-fly during rasterization • Analytical clipping: alter input geometry
Today: Line Clipping & Rasterization
Modeling Transformations Illumination (Shading) Viewing Transformation (Perspective / Orthographic) Clipping Projection (to Screen Space) Scan Conversion (Rasterization) Visibility / Display
– pass through
n
• If H•p < 0 and H•q < 0
q
MIT EECS 6.837, Durand and Cutler
Segment Clipping
• If H•p > 0 and H•q < 0
– clip q to plane
p
• If H•p < 0 and H•q > 0
Segment Clipping
• If H•p > 0 and H•q < 0
– clip q to plane
• If H•p < 0 and H•q > 0
– clip p to plane
n
• If H•p > 0 and H•q > 0 • If H•p < 0 and H•q < 0
p
Result is a single segment. Why?
MIT EECS 6.837, Durand and Cutler
Line – Plane Intersection
• Explicit (Parametric) Line Equation
L(t) = P0 + t * (P1 – P0) L(t) = (1-t) * P0 + t * P1
Segment Clipping
• If H•p > 0 and H•q < 0
– clip q to plane
p n
• If H•p < 0 and H•q > 0 • If H•p > 0 and H•q > 0 • If H•p < 0 and H•q < 0