Geometrys are usually created from polygons. The edges of these polygons are
straight lines connecting up the vertex coordinates. How do we make curved,
organic looking surfaces, normals help to reduce the look of the sharp edges
but do not change the angular shape of the outline.
One way to do this is to generate curved surfaces mathematically by using the
vertices as control points. There are a number of methods to do this as described
in the following sections:
Another use for this type of curve fitting is in animation, where we want to
change some variable such as position over time.
Bezier Curves
(cubic) Bezier curves can be generated recursively by joining up the midpoints
of the lines joining the midpoints. The equation of this curve is given by:
or
B(t) = (1-t)^3 p0 + 3t(1-t)^2 p1 + 3t^2(1-t) p2 + t^3 p3
B-Spline
The generated curve does not necessarily pass through the control points, however
if B-splines are calculated for adjacent sets of control points the curve segments
will join up and produce a continuous surface. The equation for B-spline is:
or
B(t) = 1/6 (-p0 + 3p1 -3p2+ p3)t^3 + 1/2 (p0 - 2p1 + p2) t^2 + 1/2 (-p0 + p2)
t + 1/6 (p0 +4p1 + p2)
NURBS (Non-Uniform Rational B-Spline)
This allows a curvy surface to be rendered directly, without first generating
a tessellated surface. This uses control methods called evaluators. NURBS is
supported by OpenGL.
Modeling splines using Java
I don't know of any general purpose spline tools in Java, but there are spline
tools for specific tasks:
If you want to draw a cubic curve in 2D you could use java.awt.geom.cubicCurve2D
If you want to move thorough 3D space using a splineInterpolator using Java3d
then you could use the following utilities:
com.sun.j3d.utils.behaviors.interpolators.RotPosScaleTCBSplinePathInterpolator
com.sun.j3d.utils.behaviors.interpolators.TCBKeyFrame
first you need a set of keyframes:
TCBKeyFrame[] splineKeyFrames = new TCBKeyFrame[nb];
for (int i = 0; i < nb; i++) {
splineKeyFrames[i] = new TCBKeyFrame((float)i/(nb - 1), 0, knots[i],r,
s, 0.0f, 0.0f, 0.0f);
}The you can then put this into an interpolator and put this into the scenegraph:
TransformGroup target = new TransformGroup();
RotPosScaleTCBSplinePathInterpolator splineInterpolator =
new RotPosScaleTCBSplinePathInterpolator(myAlpha, target, new Transform3D(), splineKeyFrames);
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| Correspondence about this page |
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Book Shop - Further reading.
Where I can, I have put links to Amazon for books that are relevant to
the subject, click on the appropriate country flag to get more details
of the book or to buy it from them.
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Curves and Surfaces in Geometric Modeling: Theory and Algorithms
The NURBs Book (Monographs in Visual Communication) - A NURBS book for software
developer, it covers the theory and also the practical side.
Other Books about Curves and Surfaces
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Commercial Software Shop
Where I can, I have put links to Amazon for commercial software, not
directly related to the software project, but related to the subject being
discussed, click on the appropriate country flag to get more details of
the software or to buy it from them.
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Can you help?
Please send me any improvements to
here. I would appreciate ideas to make the pages more useful including
error correction, ideas for new pages, improvements to wording. It helps
if you quote the full URL of the page.
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progam
I am working on a project which uses these principles, if you would like
to help me with this you are welcome to join in, here:
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http://sourceforge.net/projects/mjbworld/
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