Don  Lancaster's Guru's Lair  Cubic Spline Library
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Cubic Splines ( some of which are called Bezier curves ) are the
key secret to creating smoothly flowing graphic curves. Curves
such as those used in PostScript fonts or in Nurbs solid models.

This library shelf holds tutorials and utilities on cubic spline use,
modification, and generation. Many of the files need fancy math
techniques. Especially those from author Jim Fitzsimmons.

Most library files require an Adobe Acrobat 4.0 plug in for online
viewing.  Offline, these may be accessed by an Acrobat Reader or
an eBook Reader or any other .PDF accepting program.
 

		Hardware Hacker #62 HACK62.PDF       Using cubic splines. Switchmode resources. Electrorheological      fluids. National's simple switcher. Colorized avuncular sleezoids.      Curvetracing contest. Non-ionizing radiation safety.

The math behind Cubic Splines CUBEMATH.PDF #58      A tutorial on exactly where the underlying math behind Bezier      cubic splines comes from. Includes plots of Basis Functions and      the "snake in the box" visualation.

Click here for the GuruGram library.
Click here for the Guru 58 source code.
Click here to Contact the Author directly.


Cubic Spline through four points NUBZ4PTS1.PDF #59      A tutorial and companion utility that lets you draw a Bezier Cubic      spline through four data points, all of which are on the curve.      Both "good" and iterative "shortest" solutions are shown.

Click for the PostScript 4 point utility.
Click here for the GuruGram library.
Click here for the Guru 59 source code.
Click here to Contact the Author directly.





Cubic Spline Length & Subdivision NUBZLEN1.PDF #60      A tutorial and companion utility that lets you find the length of a      Bezier Cubic Spline, as well as moving to any point on that curve,      and subdividing into linear segments.

Click for the Spline Length & Subdivide utility.
Click here for the PostScript library.
Click here for the GuruGram library.
Click here for the Guru 60 source code.
Click here to Contact the Author directly.


Circle Cubic Spline Approximation ELLIPSE4.PDF #57      Circles and ellipses can be accurately approximated by four properly     chosen cubic splines. Tutorial derives the approximation math for     various spline fits from two through eight.

Click here for the PostScript library.
Click here for the Guru 57 source code.
Click here to Contact the Author directly.


		Bezier curve recursive midpoint rule  MIDPOINT.PDF       Jim Fitzsimmons TEX tutorial shows how to approximate a Bezier      curve by successive midpoint line segments. This is also called the      Casteljau method using Bernshtein Polynomials.

Bezier curve through four points  BEZ4PTS.PDF       Jim Fitzsimmons TEX tutorial shows how to fit a Bezier curve      through two end points and two intermediate points, all of which      are on the desired cubic polynomial curve.

       Click here for a practical example of BEZ4PTS.PDF:
       Click here for a new and improved version:

Bicubic Interpolation tutorial from Michael Aramini      A tutorial on the "best" nonadaptive and fairly computationally      efficient means of resampling picture bitmaps.

Click here for our Consulting Services.
Click for our Images & Fonts library page.

Using Cubic Splines for Image Iterpolation BASIS.PDF       Cubic spline Basis Functions are a powerful but little understood       method of curve control. Tutorial includes fundamentals along with       application of hires cubic splines to image interpolationn .

Click here for the BASIS.PSL sourcecode.

Image Interp Table Lookup Generator IMTABLE.PSL       Complex cubic functions can be replaced by simple table lookups       when using Basis Functions. This example shows how to create the       PostScript needed tables for hires cubic spline image interpolation.

Click here for the IMTABLE.PDF checkplot .

Swings & Tilts for your Digital Camera! SWINGT01.PSL      A PS language .BMP file modifier that adds view camera effects to      your digital camera! Uses true Bicubic Interpolation. View in wp or      editor, change filenames & params, resave, send to Distiller.

Click here for the Before .BMP testfile.
Click here for the Before .JPG example.
Click here for the After .JPG example.

Click here for the Gonzo utilities.
Coming soon: GuruGram tutorial.

Pixel Interpolation Algorithms PIXINTPL.PDF #83      A review of popular pixel interpolation schemes including Bilinear,      Nearest Neighbor, Bilinear with lookup, Modified 3x3, and Bicubic.      Includes detailed bicubic math derivation. .

Click here for the Pixel Interpolation utility.
Click here for the Pixel Interpolation demo..

Click here for the PostScript library.
Click here for the GuruGram 83 source code.
Click here to Contact the Author directly.


		Bezier approximation to circular arc I  BEZARC1.PDF       Jim Fitzsimmons TEX tutorial shows how to use a Bezier curve      to approximate a circular arc while exactly meeting end points and      end slopes of the desired circular segment.

		Bezier approximation to circular arc II  BEZCIRC2.PDF       Richard DeVeneza PDF tutorial shows a second method of using a      Bezier curve to approximate a circular arc while exactly meeting end      points and end slopes of the desired circular segment.

		Subdividing a cubic spline  BEZCHORD.HTML       Don Lancaster tutorial and demo shows how to subdivide a cubic      spline using the "constant error", "constant t", and the "constant      chord methods. Includes Postscript-as-language demo plot.

Click here for the PostScript Sourcecode
Or here for an Acrobat PDF Demo.
 

		Fitting points to a bezier curve  BEZDIST.PDF       Jim Fitzsimmons TEX tutorial shows how to fit a Bezier curve      to a group of data points. Includes an iterative BASIC program and      several demo plots that do a least squares fit to multiple points.

		Fast Bezier curve evaluation FASTBEZ.PDF       Jim Fitzsimmons TEX tutorial shows a rapid method of     generating a bezier curve. Includes TEX sourcecode that     shows one method of showing math equations.

		Connecting two Bezier curves  BEZCONN.PDF       Jim Fitzsimmons TEX tutorial shows a way to connect two      adjacent Bezier curves and replace them with a new pair of      Bezier curves whose slope is identical at the common point.

		Hyperbolic cubic spline fitting  HYSPLINE.PDF       Don Lancaster Acrobat tutorial shows the steps involved in      approximating a 1/x hyperbola with a pair of cubic splines, done      to exceptional accuracy.

Fitting 3 points to a quadratic curve PARAFIT.PSL      Brief Don Lancaster tutorial and PostScript demo of fitting three     data points to a parabola and quickly finding the x and y values of     the minimum. For curve fitting and minimization.

Click here for additional PostScript Resources.
Or here for additional Math Tutorials.


		An alternate view of a Bezier curver BEZIERU.PDF      Jim Fitzsimmons TEX tutorial shows a "balanced" way of viewing     a Bezier curve. Instead of a "t" variable that goes from 0 to 1, a     "u" variable (u = 2t-1) is used, ranging from -1 to +1.

		Some more cubic spline math  BEZMATH.PS       Don Lancaster tutorial and demo shows how to find y values for      x values of a cubic spline. Even when loops and cubic equations     are involved. Read as tutorial, then extract PostScript utilities.

Click here for the PostScript Sourcecode
 

		Bezier curve sinewave approximation  BEZSINE.PDF       Jim Fitzsimmons TEX tutorial shows an eight step approximation      to a full sinewave that uses Bezier cubic splines. Each spline curve fits      45 degrees and exactly matches value and slope at end points.

		Accuracy of PostScript circles  BEZCIRC.PDF       Jim Fitzsimmons TEX tutorial analyzes the errors involved in a      four curveto approximation to a PostScript circle. Worst case      error is one pixel at 1200 DPI for a 230 point diameter circle.

		Bezier curve through fuzzy data!  FUZZYBEZ.HTML      Don Lancaster tutorial demo shows how to fit a Bezier curve to fuzzy    data points. While preserving end points and the exit and entry    angles. Read PS sourcecode in editor, modify then send to Distiller.

Click here for the PostScript Sourcecode
Or here for an Acrobat PDF Demo.
 

		Graphics on a Bezier Surface  GURU68.PDF       Don Lancaster column reprint that gives a summary of how to      map text and graphics onto a Bezier surface. Text on a waving      flag is used as an example.

		Length of a Bezier Curve  GURU60.PDF       Don Lancaster column reprint that gives a summary of how to      quickly approximate the length of a Bezier curve. Apparently      no simple and closed form math solution is known.

		Real length of a Bezier curve! BEZLENJF.PDF       Jim Fitzsimmons tutorial shows how to use elliptic integrals to find      the real length of a Bezier curve. For most users, approximations      will often end up faster. Advanced math skills needed.

		3 Bezier curve drawing algorithms  BEZGEN3.HTML       Don Lancaster tutorial demo shows three ways to generate a cubic      spline or Bezier curve. Includes PostScript's curveto operator, brute      force t parameters, and elegant "addition only" differencing.

Click here for the PostScript Sourcecode
Or here for an Acrobat PDF Demo.
 

		Bezier curve book review  MUSE89.PDF       Brief reviewof Knot Insertion and Deletion Algorithms for      B-Spline Curves textbook. On the tricky process of breaking up      a curve into any number of subcurves. Lots of references.

Here's a tentative collection of Bezier books.Please let me
know of any that should be added to this list...

B-Spline Knot Insertion and Deletion Algorithms ( Roger Goldman ) Curves and Surfaces  ( Academic Press ) Curve and Surface Fitting  ( Peter Lancaster ) An Introduction to Nurbs  ( David Rodgers ) Geometric Modeling using Beta Splines  ( Brian Barsky ) Handbook of Splines  ( Gheorghe Micula ) Handbook of Splines for the User  ( Eugene Shikin ) Interpolating Cubic Splines  ( Gary Knott ) Methods of Shape; Preserving Spline Approximation ( Boris Kvasov ) The Nurbs Book  ( Les Piegelr ) Nurbs for Curves and Surface Design  ( Gerald Farin ) PostScript Language Reference Manual   ( Adobe Systems ) A Practical Guide to Splines ( Carl De Booer ) Spline Functions ( K. Bohmer ) Theory of Splines and Applications  ( J. Ahlberg ) Two Dimensional Spline Interpolation Algorithms  ( Helmuth Spath )

Here are a few worthwhile Bezier and Cubic Spline links that you
might like to check into...

http://www.sunsite.ubc.ca/LivingMathematicsl http://www.moshplant.com/direct-or/bezier http://theory.lcs.mit.edu/~boyko/classes/bezier-spline.html http://www.gamasutra.com/features/19991027/deloura_01.htm http://www.ldraw.org/memorial/archive/bezier.html http://www.cs.huji.ac.il/~arik/java/ex2/ http://cgm.cs.mcgill.ca/~luc/bezier.html http://www.webreference.com/dlab/9902/bezier.html http://www.cs.princeton.edu/~min/cs426/jar/bezier.html http://www.research.microsoft.com/~hollasch/cgindex/curves/ http://www.cnet.com/Resources/Info/Glossary/Terms http://theoryx5.uwinnipeg.ca/CPAN/data/Math-Bezier http://www.cee.hw.ac.uk/~ian/hyper00/curvesurf/bezier.html

And here are some possible newsgroups that sometimes have
useful Bezier or Cubic Spline content...

alt.math comp.graphics comp.graphics.algorithms comp.graphics.misc comp.graphics.rendering comp.lang.postscript comp.soft-sys.math sci.math sci.math.research

Ask Don a Cubic Spline or Bezier question.   Place a Cubic Spline Consulting Order.  Find Instant Research Solutions.

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