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SplineLibrary

A C++ library created to provide open-source reference implementations of many spline functions, e.g. Natural Splines and Catmull-Rom Splines. It's a challenge to find well-documented and FOSS implementations of these useful tools, and mathematical defintions of these things are really hard to decipher, so my hope is that this library will provide a starting point for others.

A spline is a formula for smoothly transitioning from one data point to the next in a data set. For example, you could create a spline containing ten colors (each stored as R, G, and B values) to create a color gradient that smoothly transitions from one color to the next.

Example

Create a Catmull-Rom Spline with four data points, and compute the interpolated position at T = 0.5

std::vector<QVector2D> splinePoints{
    QVector2D( 0, 0),
    QVector2D( 5, 0),
    QVector2D( 8, 3),
    QVector2D( 6, 1),
};
UniformCRSpline<QVector2D> mySpline(splinePoints);
QVector2D interpolatedPosition = mySpline.getPosition(0.5f);

Features

• Interpolation of catmull-rom splines
  • Include spline_library/splines/uniform_cr_spline.h, create a UniformCRSpline object, and call its getPosition method.
  • Several more spline types. See Spline Types for the full list
• Looping Splines
  • Also called "Periodic" or "Cyclic": These splines form a loop, in that the ending connects with the beginning
  • Calling getPosition(t) with an out-of-range T value will "wrap around" to the other end of the spline
  • To make a looping catmull-rom spline, include spline_library/splines/uniform_cr_spline.h and create a LoopingUniformCRSpline object.
  • Every spline type has both looping and non-looping variants
• Compute the arc length of a spline
  • Call a spline's totalLength() method to find the arc length of the entire spline
  • Call a spline's arcLength(a,b) method to find the arc length between two arbitrary T values
• Compute the inverse of a spline
  • Given a data point (not necessarily on the spline, or even close to it), what T value brings the spline closest to that data point?
  • Create a SplineInverter object and call its findClosestT method
• Computation of the first, second, and third derivatives of the spline
  • The first derivative is called the "tangent" - this is how quickly and in what direction the interpolated position is changing, per T
  • The second derivative is called the "curvature" - this is how quickly and in what direction the interpolated tangent is changing, per T
  • The third derivative is called the "wiggle" - this is how quickly and in what direction the interpolated curvature is changing, per T

Documentation

Glossary - Glossary of important terms for understanding splines.

Spline class API - API documentation of the Spline base class.

Spline Types - Complete list of all supported spline formulas

Spline Utilities - Documentation of some utility classes for splines.

Project Layout

The root of the repository is a Qt Creator project that demonstrates some uses of the library. The source for the spline code itself is in the "spline_library" directory, and the code to set up the demo is in the "demo" directory.

Usage

Drop the spline_library directory in the root source folder of your project. It's header-only, so from here all you need to do is import it from your own code.

spline_library/spline_inverter.h and spline_library/arclength.h depend on Boost's Math module. If you don't want to install Boost, you can safely avoid including these two files - nothing else includes it, and nothing else relies on Boost.

Both the demo and the spline_library code require a fully compliant C++14 compiler.

Demo

Follow these steps to run the demo (Assuming you already have Qt 5.5+ installed and working):

1. Install Boost. On Linux, this is in most package managers. On Mac, it can be installed via homebrew. Otherwise, visit http://www.boost.org/

2. Create a file called SplineDemo_Include.pri in the root of the project

3. In this file, paste the following, where /path/to/boost contains Boost's include files.

   INCLUDEPATH += "/path/to/boost"

   On windows, this might be: INCLUDEPATH += "C:\Boost\boost_1_60_0"

   On Mac, this might be: INCLUDEPATH += /usr/local/Cellar/boost/1.59.0/include

4. Run qmake on SplineDemo.pro to generate a makefile, then build the makefile, and run the compiled executable

5. OR, open SplineDemo.pro in Qt Creator and press play

License

This code is available under the Simplified BSD License

This project includes the nanoflann library for fast nearest-neighbor queries, which is also available under the Simplified BSD License

To-Do

• Implement "composite splines", ie a spline that is made by combining two or more splines. Current ideas include a "sum spline" where the output for a given T is a sum of all the child spline results at T, and a "concatenation spline" formed simply by starting one spline where the previous leaves off.
• More spline types as I discover them
• Find an actual mathematical definition for the quintic catmull-rom spline. The quintic cubic hermite spline is well-defined, but I basically guessed on how to automatically compute the tangents and curvatures based on the input points for the catmull-rom equivalent.