1. COMPUTER GRAPHICS CHAPTERS 22-23 CS 482 – Fall 2017 SPLINES
CUBIC CURVES
HERMITE CURVES
BéZIER CURVES
B-SPLINES
BICUBIC SURFACES
SUBDIVISION SURFACES

2. CUBIC CURVES LINEAR LIMITATIONS
Straight line segments may be used to approximate curves, but...
Either the approximation has too many sharp angles, or...
…the number of line segments required is prohibitively large.
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3. CUBIC CURVES PARAMETERIZED CUBICS
Cubics are the lowest order polynomials capable of illustrating maxima, minima, concavity, and inflection points.
Notice that each equation has four unknowns, so four pieces of information would define each cubic equation!
By defining the cubic curve parametrically, restricting the parameter t to the [0,1] interval, we obtain a concise structure to contain a good approximation to a desired curve.
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4. HERMITE CURVES ENDPOINTS & TANGENT VECTORS
The Hermite form of the cubic polynomial curve is constrained by the endpoints (P0 and P1) and the tangent vectors at the endpoints (P0´ and P1´).
P0´ P0 P1
P1´
Similar solutions for the y- and z-coordinates yield:
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5. HERMITE CURVES EXAMPLES
The big problem with the Hermite approach is the requirement that the tangent vectors at the endpoints must be specified in advance.
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6. BéZIER CURVES ENDPOINTS & CONTROL POINTS P2
The Bézier form of the cubic polynomial curve indirectly specifies the tangent vectors at endpoints P1 and P4 by specifying two intermediate points (P2 and P3) that are not on the curve. P2 P4 P1 P3
Calculations similar to those derived for the Hermite form yield:
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7. BéZIER CURVES FIRST-ORDER DISCONTINUITY
The Bézier form has zero-order continuity, but it lacks first-order continuity (unless the triple of vertices around the “knot” happen to be collinear).
Discontinuous First Derivative P7
Discontinuous First Derivative P5 P4 P8 P6 P2 P3 P9
P10 P1
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8. B-SPLINES CONTROL POINTS ONLY
To ensure first-order (and even second-order) continuity at the “knots” adjoining consecutive cubic curve segments, a B-spline approach is taken, with the drawback that the curve passes through none of the control points. P2 P4 P1 P3
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9. SECOND-ORDER CONTINUITY
B-SPLINES
SECOND-ORDER CONTINUITY P7 P5 P4 P8 P6 P2 P3 P9
P11
P10 P1
The B-spline knots have first-order continuity (i.e., smooth tangents) and second-order continuity (i.e., smooth concavity), but require three times as many parameterizations. P0
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10. BICUBIC SURFACES APPROXIMATING SMOOTH SURFACES
Line segments may be used to approximate surface boundaries, but...
100 Triangles
1000 Triangles
69,451 Triangles
Determining the segment endpoint coordinates is a big job!
It takes a lot of triangles to yield a decent image, even with shading!
The resulting storage and processing costs could be prohibitive!
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11. BICUBIC SURFACES PATCHES
Patches made up of orthogonal cubic curves can be used to approximate surfaces.
Hermite patches require partial derivatives at every vertex.
Bézier patches require collinearity between adjacent patches.
B-spline patches require nine times as many parameterizations.
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12. Subdivided Bézier Curves
SUBDIVISION SURFACES
USING BEZIER PATCHES
Patches can be refined by creating a larger grid of control points from the original grid of control points.
Original Bézier Curve P3
L1 = P1
R4 = P4 P2 R2
L4=R1 L3
L2 = ½(P1+P2)
R3 = ½(P3+P4) R3 L2
L3 = ½(L2+½(P2+P3))
R2 = ½(R3+½(P2+P3))
Subdivided Bézier Curves P4 R4
L4 = R1 = ½(L3+R2) L1 P1
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13. SUBDIVISION SURFACES APPLICATIONS
Reverse engineering on sparse data (e.g., limited medical scans)
Controlling surface quality according to needs and processing abilities
Transmission and compression of 3D mesh data
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