lecture14 - CSE472 Computer Graphics Curves I Curves...

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1 CSE472 Computer Graphics Curves I Curves Representations of Curves and Surfaces Hermite Curves Bezier Curves Notice: A program called curves is available On class web site You can run this and try different curve ideas You can also examine the source to see how it works.
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2 CSE472 Computer Graphics Discussion Why Curves? Lines and surfaces… Why not just tessellate?
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3 CSE472 Computer Graphics Curve advantages Scale independence You can tessellate when you know the actual display scale Natural description Great for simulating continuous motion! Curves between fixed control points…
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4 CSE472 Computer Graphics How would you describe this?
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5 CSE472 Computer Graphics Explicit Representation y = x 2 Explicit representation represents a curve with one variable as a function of one or more other variables. y=f(x)
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6 CSE472 Computer Graphics Explicit Representation 2D y = f(x) (or x = f(y)) 3D z = f(x, y) 2D Line y = mx + b What’s the representation for a 3D line?
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7 CSE472 Computer Graphics Explicit falls apart 3D Line requires two equations y = f(x) z = f(x) But, what about a line in a plane parallel to yz? What about a circle? y = sqrt(r 2 – x 2 )
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8 CSE472 Computer Graphics Implicit Representation x 2 - y = 0 Implicit representation represents a curve as a function of all variables equal zero: f(x, y) = 0
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9 CSE472 Computer Graphics Implicit Representation 2D f(x, y) = 0 ax +by + c = 0 Line x 2 + y 2 – r 2 = 0 Circle 3D f(x, y, z) = 0 ax + by + cz +d = 0 Plane x 2 + y 2 + z 2 – r 2 = 0 Sphere
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10 CSE472 Computer Graphics Problems with Implicit Form The function is really a membership test f(x, y) = 0 Does f(x,y) = 0 for a point? Impractical to test points
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11 CSE472 Computer Graphics Parametric Representation We introduce a new, artificial parameter One equation for each dimension x = x (u), y = y (u) Example: y = u 2 x = u Also called: parametric form x 2 - y = 0
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12 CSE472 Computer Graphics Example: A Line Let ax + by + c = 0 describe a line. x = -cu/a y = c(u-1)/b (Unless horizontal or vertical) Vertical (ax + c = 0): x = -c/a, y = u Horizontal (you figure it out…)
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13 CSE472 Computer Graphics Don’t believe me? ax + by + c = 0 describes the line I said: x = -cu/a y = c(u-1)/b Plug them in: a (-cu / a) + b(c(u-1)/b) + c = 0 -cu + cu – c + c = 0 How twoo, how twoo, how twoo…
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CSE472 Computer Graphics What does u represent? For a line:
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lecture14 - CSE472 Computer Graphics Curves I Curves...

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