Bezier spline |
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Some more examples:
Definition of a Bezier curve (de Casteljau, 1959; Bezier, 1962)
Let's start simple. Imagine a line that begins at A and ends at D. Let t be a variable from 0 to 1. We can define a point PAD(t) on the line segment AD as
PAD(t) = (1-t)A + tD
If we add another point, B, into the picture, we can define PAB(t) as a point between A and B, and PBD(t) as a point between B and D. If we apply the same method to define PAB-BD(t) as a point between PAB(t) and PBD(t), we get
PAB(t) = (1-t)A + tB PBD(t) = (1-t)B + tD PAB-BD(t) = (1-t){ (1-t)A + tB } + t{ (1-t)B + tD) } = (1-t)2A + 2t(1-t)B + t2D
This quadratic equation in t defines a quadratic Bezier curve - a parabola. For computer graphics purposes, cubic Bezier curves are more often used. As you might expect, these are defined by four points A, B, C, and D. If the cubic Bezier function is defined as a point between PAB-BC(t) and PBC-CD(t) in the same manner as we've been doing so far...
PAB(t) = (1-t)A + tB PBC(t) = (1-t)B + tC PCD(t) = (1-t)C + tD PAB-BC(t) = (1-t){ (1-t)A + tB } + t{ (1-t)B + tC) } = (1-t)2A + 2t(1-t)B + t2C PBC-CD(t) = (1-t){ (1-t)B + tC } + t{ (1-t)C + tD) } = (1-t)2B + 2t(1-t)C + t2D CubicBezier(t) = (1-t)PAB-BC(t) + tPBC-CD(t) = ... = (1-t)3A + 3(1-t)2tB + 3(1-t)t2C + t3D
Summary: Where we end up
Cubic Bezier splines are usually defined with endpoints A and D and control points B and C that are not on the curve, as above. The equation for a point on this curve is given by
CubicBezier(t) = (1-t)3A + 3(1-t)2tB + 3(1-t)t2C + t3D
where t is the curve's parameter and ranges from 0 to 1. This curve can be expressed in a different way: as a curve passing through four points, PQRS, where P=CubicBezier(0), Q=CubicBezier(1/3), R=CubicBezier(2/3), and S=CubicBezier(1). From the formula above,
P = A Q = 1/27(8A + 12B + 6C + D) R = 1/27(A + 6B + 12C + 8D) S = D
and therefore
A = P B = 1/6(-5P + 18Q - 9R + 2S) C = 1/6(2P - 9Q +18R - 5S) D = S
In RatBat's Bezier dialogue box (above), you enter the points ABCD and RatBat shows you the points PQRS ("The curve passes through these points: ...").
Some properties of Bezier curves
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