Transcript (Hermite) Curves

(HERMITE) CURVES
Reference: http://en.wikipedia.org/wiki/Cubic_Hermite_spline
Lecture #4
BASIC IDEA
 [Adobe Illustrator / Inkscape demo]
 [Uses]
 A dynamic curve defined by:
Hermite Curve
 Control Points (Points)
 Tangents (Vectors)
 Hermite Curves are one of the simpler
curves. Others include:
 Bezier
 NURBS (Non-uniform rational Basis Spline)
…
 Hermite curves (unlike some) actually pass
through the control points.
 NURBS, for example, don’t.
 Options:
 Allow discontinuities? [2 tangents per C.P.]
 Loop? [Last C.P. & Normal = First]
NURBS Curve
CALCULATING (DRAW) POINTS
 Suppose we have a curve segment defined by:
 2 control points (𝑝0 is the start, 𝑝1 is the end)
 2 tangent directions (𝑡 0 at the start point, 𝑡1 at the end point)
 This is a vector (it might be calculated from a tangent “handle”: a
position)
 Curves are usually defined with a parametric formula
 Involves a parameter, u, which goes from 0.0 (at the start) to 1.0 (at
the end)
 We can generate intermediate points using this formula:
𝑝(𝑢) = 𝑎 𝑝 0 + 𝑏𝑡 0 + 𝑐 𝑝1 + 𝑑 𝑡 1
𝑎 = 2𝑢 3 − 3𝑢 2 + 1
𝑏 = 𝑢 3 − 2𝑢 2 + 𝑢
𝑐 = −2𝑢 3 + 3𝑢 2
𝑑 = 𝑢3 − 𝑢2
A COMPLETE CURVE
 Just have 0 – n control points (and a matching tangent)
 When generating the complete curve
 Look segment-by-segment (each segment has a 0 <= t <= 1)
 If we have n control points, there are (n-1) segments.
 DrawList




In pygame it is handy to store a “draw list” of generated points
Perhaps controlled by a “resolution” variable?
Just use pygame.draw.line to connect them
Done!