260

# SER431 Lecture 13

Curves and Splines
(201810)

October 01, 2018

## Transcript

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SER 431
Lecture 13: Curves, Splines, and Surfaces
Javier Gonzalez-Sanchez
[email protected]
PERALTA 230U
Office Hours: By appointment

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Surfaces

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Surfaces
Can be represented by huge number of points (polygons or triangles)
+ arbitrary shapes possible
– large memory requirements
– changes cause much work
– corners
- scaling
Can be represented by two sets of orthogonal curves
– only for some shape categories
+ marginal memory requirements
+ changes are rather simple
+ definition arbitrarily exact
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Curves | Concept
A curve is
a generalization of a line (a set of points),

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Curves | Concept
A curve is
a generalization of a line (a set of points),
in that its curvature need not be zero.

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Curves | Representation
§ Implicit curve. y = f(x)
such as y = sqrt (1–x2)
§ Parametric curve. x = f(t), y = g(t)
such as x=cos(t), y=sin(t)

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• Derivatives of the coordinates define
the tangent
• The length of the tangent does not
have a geometric meaning. Consider
only +, 0, -, and ∞
• From slope
• To “rate of change” at any point
(first derivative)
Curves | First Derivative

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• Derivatives of the coordinates define
the tangent
• The length of the tangent does not
have a geometric meaning. Consider
only +, 0, -, and ∞
• From slope
• To “rate of change” at any point
(first derivative)
Curves | First Derivative

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Curves | First Derivative
The curvature is the inverse of the radius of the best local approximation of
the curve by a circle.
And, curvature is defined as the magnitude of the derivative of a unit
tangent vector function with respect to arc length:
Moreover, for small values curvature is approximately the second derivative.

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Curves | Second Derivative
The second derivative measures the concavity
§ positive mean convex, i.e., tangent line below the graph (blue)
§ zero represents an inflection point (red)
§ negative mean concave, i.e., tangent line above the graph (green)

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Spline Functions
A curve can be defined piecewise by polynomials (spline functions)
§ Interpolating splines
§ Approximating splines
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Curves and Splines
§ A curve can be a concatenation of splines.
§ Control points determine the shape of the spline curve.
§ For each point specify a blending function which determines how the
control point influence the shape of the curve for values of parameter t.
§ Therefore, a curve spline is specified as
P(t) = p0
*B0
(t) + p1
*B1
(t) + p2
*B2
(t) + p3
*B3
(t) + ...
§ This is axis independent, i.e., it does not change when the coordinate
system is rotated.
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§ A control graph (or control polygon) is a poly-line connecting control
points.
§ It shows the order of control points
Control Graph
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Blending functions | Example Bezier
Blending function are:
§ Easy to compute (polynomials are)
§ Continuous
§ Interpolate nicely the control points
Example (Bezier blending function):
§ Define 4 control points
§ Define t between 0 and 1
§ Blending function are defined as
B0
(t)=(1-t)3, B1
(t)=3t(1-t)2, B2
(t)=3t2(1-t), B1
(t)=(t)3

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Blending functions | Example Bezier
Blending function are:
§ Easy to compute (polynomials are)
§ Continuous
§ Interpolate nicely the control points
Example (Bezier blending function):
§ Define 4 control points
§ Define t between 0 and 1
§ Blending function are defined as
B0
(t)=(1-t)3, B1
(t)=3t(1-t)2, B2
(t)=3t2(1-t), B1
(t)=(t)3

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Thus, with 4 points and the Bezier blending functions, we replace
• P(t) = p0
*B0
(t) + p1
*B1
(t) + p2
*B2
(t) + p3
*B3
(t)
With
P(t) = (1-t)3 P0
+ 3t(1-t)2 P1
+ 3t2(1-t) P2
+ (t)3 P3
Where t is 0 <= t < = 1

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Source Code

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Step 1
// 4 control points
float Points[4][3] = {
{ 10, 0, 0 },
{ 5, 10, 2 },
{ -5, 0, 0 },
{-10, 5, -2 }
};
P3
P0
P1
P2

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Step 2
// draw the control vertices in red.
// It is only as a reference. They are not need on screen. Ok?
glColor3f(1, 0, 0);
glPointSize(3);
glBegin(GL_POINTS);
for (int i = 0; i != 4; ++i) {
glVertex3fv(Points[i]);
}
glEnd();
P3
P0
P1
P2

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Step 3
// control graph in blue
glColor3f(0, 0, 1);
glBegin(GL_LINE_STRIP);
for (int i = 0; i != 4; ++i) {
glVertex3fv(Points[i]);
}
glEnd();
P3
P0
P1
P2

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Step 4
glBegin(GL_LINE_STRIP);
glColor3f(0, 1, 0);
// N = 20
for (int i = 0; i != N; ++i) {
float t = (float) i / (N - 1); // 20 values 0 to 1
float it = 1.0f - t;
// blending functions
float b0 = t*t*t; //(t)3 for P3
float b1 = 3 * t * t * it; // 3t2(1-t) for P2
float b2 = 3 * t * it*it; // 3t(1-t)2 for P1
float b3 = it*it*it; //(1-t)3 for P0
// P(t) = (1-t)3 P0 + 3t(1-t)2 P1 + 3t2(1-t) P2 + (t)3 P3
float x = b3 * Points[0][0] + b2 * Points[1][0] + b1 * Points[2][0] + b0 * Points[3][0];
float y = b3 * Points[0][1] + b2 * Points[1][1] + b1 * Points[2][1] + b0 * Points[3][1];
float z = b3 * Points[0][2] + b2 * Points[1][2] + b1 * Points[2][2] + b0 * Points[3][2];
// specify the point
glVertex3f(x, y, z);
}
glEnd();

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Examples
N = 20 N = 5
N = 2

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§ Review the source code posted on GitHub
Test yourselves:
§ Change the coordinates of the control points; for instance, play with diverse
combinations of positive and negative x and y.
§ Draw a second Bezier spline adjacent to the first one
§ Draw a second Bezier spline parallel to the first one
Homework

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