Topics in Applied Mathematics
Douglas Wilhelm Harder
Department of Electrical and Computer Engineering
University of Waterloo
Waterloo, Ontario, Canada
Copyright © 2009 by Douglas Wilhelm Harder. All rights reserved.
Best-Fitting Bezier Curvesfor Graphs of Functions
Best Fitting Bezier Curves
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Outline
• This presentation looks at finding optimal Bezier curves– Background and definitions– Eight unknowns:
• Six constraints based on the functions and derivatives
• Two constraints based on minimizing the mean squared error
– Ten examples of various functions
Best Fitting Bezier Curves
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Background
• Bezier curves are used to model smooth space curves• Often necessary to plot specific functions, including:
– Circles– Exponential– Trigonometric
functions• We will define a technique for finding such curves and
look at some examples
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Mathematical Notation
• We will use standard mathematical notation to represent various values:– Variables are italicized lower-case characters t, x, y, a, b
– Vectors are bold lower-case characters vk
– Functions are roman lower-case characters f(t), x(t), y(t)
– Derivatives are represented as superscripts f(1)(t)
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Theory
• Given a vector space, a cubic Bezier curve is defined by four vectors
v1, v2, v3, v4
and uses these to define a space curve parametrically
v1(1 – t)3 + v2t (1 – t)2 + v3t 2(1 – t) + v4t3
with t varying from 0 to 1
• We will use vk = (xk, yk)
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Theory
• To find the best fitting Bezier curve for the graph of a function (x, f(x)) for x on an interval [a, b], introduce three constraints:
v1 = (a, f(a))
v4 = (b, f(b))
• We will also require that the derivatives match:
f(1)(a) = (y2 – y1)/(x2 – x1)
f(1)(b) = (y4 – y3)/(x4 – x3)
where vk = (xk, yk)T
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Theory
• To deal with functions which have asymptotes in the derivative, to constrain the size, we will make one of two substitutions: – If |f(1)(a)| < 1, substitute y2 = f(a) + f(1)(a) (x2 – a),
else substitute x2 = (y2 – f(a))/f(1)(a) + a
– Similarly, if |f(1)(b)| < 1, substitute y3 = f(b) + f(1)(b) (x3 – b),
else substitute x3 = (y3 – f(b))/f(1)(b) + b
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Theory
• With these substitutions, we now have two Bezier curves with two unknown variables– These define the Bezier curve parametrically
• There are four cases for the substitutions:– If both derivatives are small, the two curves are
x(t) = a(1 – t)3 + x2t (1 – t)2 + x3t 2(1 – t) + bt3
y(t) = f(a)(1 – t)3 + [f(a) + f(1)(a) (x2 – a) ]t (1 – t)2 +
[f(b) + f(1)(b) (x3 – b)]t 2(1 – t) + f(b)t3
– If the derivative is large (or infinity) at the left point, use:
x(t) = a(1 – t)3 + [(y2 – f(a))/f(1)(a) + a]t (1 – t)2 + x3t 2(1 – t) + bt3
y(t) = f(a)(1 – t)3 + y2t (1 – t)2 + [f(b) + f(1)(b) (x3 – b)]t 2(1 – t) +
f(b)t3
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Theory
• We must now define a function which must be minimized• We will minimize the root mean-squared error:
• This is a real-valued bivariate function in the two unknown parameters
• Use techniques such as gradient descent to find the minimum
• Reasonable initial approximations are the midpoints:xk = (a + b)/2 yk = (f(a) + f(b))/2
tttt dxxfy 11
0
2
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Theory
• Why minimize the root mean squared error?– The error is a continuous and differentiable function in the
parameters– Squaring emphasizes larger errors relative to smaller errors
• One could also minimize the infinity norm:
• Continuous, but not differentiable
tt xfy
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Examples
• The following ten slides gives the best fitting Bezier curves for a number of functions on specific intervals– The function is plotted in red– The best fitting Bezier curve is plotted in blue behind the graph
of the function
• The code was written in the Maple programming language
• The source code is available athttp://ece.uwaterloo.ca/~dwharder/Maplesque/Bezier/
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Examples
• The best fitting Bezier curve for the quarter circle is defined by the points
(0, 1), (0.5520, 1), (1, 0.5520), (1, 0)
with a root mean squared error of 0.000175
21 xy
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Examples
• The best fitting Bezier curve for the sin function on[0, /2] is defined by the points
(0, 0), (0.5251, 0.5251), (1.005, 1), (1.571, 1)
with a root mean squared error of 0.000474
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Examples
• The best fitting Bezier curve for the exponential function on [0, 1] is defined by the points
(0, 1), (0.3765, 1.3765), (0.7057, 1.9184), (1, 2.7183)
with a root mean squared error of 0.00000102
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Examples
• The best fitting Bezier curve for the natural logarithm function on [1, 2] is defined by the points
(0.5, -0.6931), (0.8043, -0.08448), (1.3010, 0.3437), (2, 0.6931)
with a root mean squared error of 0.000238
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Examples
• The best fitting Bezier curve for the quadratic function on[0, 1] is defined by the points
(0, 0), (0.4174, 0), (0.7629, 0.5257), (1, 1)
with a root mean squared error of 0.000118
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Examples
• The best fitting Bezier curve for the square root function (quadratic inverse) on [0, 1] is defined by the points
(0, 0), (0, 0.4182), (0.5274, 0.7637), (1, 1)
with a root mean squared error of 0.000124
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Examples
• The best fitting Bezier curve for the function f(x) = x3/2 on[0, 1] is defined by the points
(0, 0), (0.1804, 0), (0.5792, 0.3689), (1,1)
with a root mean squared error of 0.00251
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Examples
• The best fitting Bezier curve for the function f(x) = x ln(x) on [1, 10] is defined by the points
(1, 0), (3.0993, 2.0993), (6.5721, 11.7052), (10., 23.0259)
with a root mean squared error of 0.0202
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Examples
• The best fitting Bezier curve for the standard normal curve on [0, 3] is defined by the points
(0,1), (0.9235, 1), (0.1722, 0.002217), (3, 0.0001234)
with a root mean squared error of 0.0490
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Examples
• It is better to use two Bezier curves on [0, 1] and [1, 3]:(0, 0.5642), (0.3452, 0.5642), (0.6314, 0.3605), (1, 0.2076)
(1, 0.2076), (1.5128, -0.005325), (1.8960, 0.0005308), (3, 0.00006963)
with a root mean squared errors of 0.0000147 and 0.000227
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Examples
• The best fitting Bezier curve for the sinc function (sin(x)/x or the cardinal sine) on [0, ] is defined by the points
(0,1), (1.1339, 1), (2.0493, 0.3477), (3.1416, 0)
with a root mean squared error of 0.000154
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Summary
• This presentation looked at finding optimal Bezier curves for function graphs:– Six constraints are defined by the properties of the graph– Two are found by minimizing the root mean squared error
• The results are usually very good– The function x1.5 resulted in the worst case– The smooth exponential function was best fit
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Usage Notes
• These slides are made publicly available on the web for anyone to use
• If you choose to use them, or a part thereof, for a course at another institution, I ask only three things:
– that you inform me that you are using the slides,– that you acknowledge my work, and– that you alert me of any mistakes which I made or changes which you make, and
allow me the option of incorporating such changes (with an acknowledgment) in my set of slides
Sincerely,
Douglas Wilhelm Harder, MMath
