Spline Morphing. CG software practical course in the IWR. Carl Friedrich Bolz. Carl Friedrich Bolz

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1 Spline Morphing CG software practical course in the IWR

2 Introduction Motivation: Splines are an important part n of vector graphics, 3D-graphics, CAD/CAM,... Splines are often used to describe characters, most font formats are based on splines Due to their properties splines are easy to morph Aims of the project: Creation of a system that allowes to create animations based on splines Simple usage Picture are rendered using POV-Ray (Persistence of Vision Raytracer)

3 Splines Three types of curves are called splines: Bézier curves B-Splines NURBS (Non-uniform rational B-Splines) All three are based on piecewise polynomial functions A control polygon is used to determine the shape of the splines How splines are used in POV-Ray: POV-Ray has splines-primitives but those can lack flexibility Therefore: splines are made out of lots of spheres

4 Bézier Curves After Pierre Bézier Polynomial curves: The points form the control polygon of the Bézier curve Points on the Bézier curve are calculated using de Casteljeau's algorithm Properties of Bézier curves The curve passes through the endpoints of the control polygon The curve lies within the convex hull of the control polygon Affine invariance The first and last line of the control polygon are tangents to the curve Cubic Bézier curves are used very often: Four control points, the curve passes through the first and the last The two other points determine the tangents

5 Example (1) De Casteljeaus algorithm:

6 Composite Bézier curves There are some problems with Bézier curve: Every point of the control polygon influences the shape of the whole curve A lot of calculation is necessary if the control polygon has many points Therefore: composite Bézier-Kurven: Several Cubic Bézier curves are used to make a larger curve The last point of the one Bézier curve is the first point of the next one If the transition between two of the Bézier curves is supposed two be smooth the points next to the common point of the two curves have to lie on a line

7 Examples (2)

8 User Interface An important aim was to make the program easy to use Since GUI programming is not always fun: An existing vector graphics program is used as a frontend Skencil ( Open Source (GPL) Written in Python and C Simple, textual file format: a Skencil file is always legal Python code (without loops, functions, etc) so Python itself can do the parsing Nearly all of Skencil's objects are supported (exception: Circle) A lot of information is taken from Skencil: color, line thickness, etc Start- and endspline are taken from the layers start end end

9 Morphing Algorithm (1) Hardest part of the project: What's looking good? Initalizing : Startsplines and endsplines are each merged into one single spline Greedy algorithm Inserted control points are marked as invisible Since start- and endspline need to have the same number of points in the control polygon they are inserted as needed The control points of start- and endspline are matched Movie: For each frame the matched control points are interpolated linearly to get the control points of the spline of the current frame Control points contain more information than just position: color, radius, visibility

10 Example for the Algorithm Startspline Endsplines

11 Morphing-Algorithm (2) Sometimes more control is needed: In addition to start- and endspline a middlespline is introduced The algorithm mostly stays the same: The splines are connected, points are inserted as needed into the control polygon, the points of the control polygons are matched Differences: Instead of a linear interpolation of the control polygons over time Bézier curve of third degree is used The control polygon of this Bézier curve is made up of the three matched points of the control polygons of start-, middle-, endspline

12 Some facts about the program The program is written completely in Python: This allowed a very fast developpment due to the very high level constructs of the language No code-compile-run-cycle Speed wasn't a problem since most of the time is spend in POV-Ray anyway A little more than 1000 lines of code Rendering a typical animation (300 frames) takes about an hour (Athlon XP 2000+)

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