Unformatted text preview: ues of 0 or 1 for µ and/or ⌫ we can easily verify
Substituting values of 0 or 1 for µ
that the equation ﬁts the edge curves. and/or ν we can easily
verify that the equation ﬁts the edge curves. !
!
Graphics Lecture 12: Slide 19!
19 / 35 Rendering a patch: Polygonisation Rendering a patch: Polygonisation "
To render (draw) a spline patch we can simply transform it
i render (draw) !
Tonto polygons. a spline patch we can simply transform it into
polygons. We select a grid of points, e.g.: !
We select a grid of points, eg:
µ = {0.0, 0.1, 0.2, . . . 1.0}
⌫ = {0.0, 0.1, 0.2, . . . 1.0} and triangulate to that grid.
and triangulate to that grid. ! Graphics Lecture 12: Slide 20!
20 / 35 Rendering a patch: Polygonisation "
Rendering a patch: Polygonisation • For speedwe can use large polygonspolygons withPhong
For speed we can use large with Gouraud or Gouraud or
shading.
Phong shading. !
• For accuracy we use small polygons, chosen to match
For accuracy we use small polygons, chosen to match the pixel size.
the pixel size. !
21 / 35 Graphics Lecture 12: Slide 21! aces can also be drawn by a technique called lofting (now
Rendering a patch: Lofting "
ly obsolete). • Surfaces can also be drawn by a technique called lofting
(now really obsolete). !
means •drawing contours of constant µ µ and/or ofof constan
and/or
This means drawing contours of constant
constant ν
• Algorithms for eliminating the hidden parts have been
rithms for eliminating the hidden parts have been devised.
devised. ! Graphics Lecture 12: Slide 22! Rendering a patch: Ray tracing Rendering a patch: Ray tracing "
The patch equation is fourth order order !
• The patch equation is fourth
P(µ, ⌫ ) = P(µ, 0)(1 ⌫ ) + P(µ, 1)⌫ + P(0, ⌫ )(1 µ) + P(1, ⌫ )µ P(0, 1)(1 µ)⌫ P(0, 0)(1 µ)(1 P(1, 0)µ(1
⌫) ⌫) P(1, 1)µ⌫ • Hence no closed form solution exists for a ray patch
intersection !
• Can numeric algorithms but computation can be can
Can use use numeric algorithms but comput...
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 Spring '14
 Surface, Parametric equation, Parametric surface

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