Spline is a piecewise polynomial curve that defines a path in a 3-dimensional space. More...
#include <kanzi/core/math/spline.hpp>
Public Types | |
| enum | { SplineDimension } |
Public Member Functions | |
| SplinePtr | clone () const |
| Clone this spline and return the cloned spline. More... | |
| const vector< float > & | getCoefficientComponents () const |
| Get the polynomial coefficients components for a spline. More... | |
| unsigned int | getDegree () const |
| Get degree of the spline polynomial. More... | |
| Vector3 | getPoint (float timePoint) const |
| Evaluates the value of the spline in the given time point. More... | |
| unsigned int | getSegmentCount () const |
| Get the spline segment count. More... | |
| bool | isLooping () const |
| Indicates whether the spline is looping. More... | |
Static Public Member Functions | |
| static SplinePtr | create (unsigned int degree, const vector< Vector3 > &coefficients, bool looping) |
| Creates a spline from polynomial coefficients. More... | |
| static SplinePtr | create (unsigned int degree, vector< float > coefficientComponents, bool looping) |
| Creates a spline from polynomial coefficients. More... | |
| static SplinePtr | createCatmullRom (vector< Vector3 > controlPoints, bool looping) |
| Creates a Catmull-Rom spline with given control points. More... | |
| static SplinePtr | createCatmullRom (const vector< float > &controlPointComponents, bool looping) |
| Creates a Catmull-Rom spline with given control points. More... | |
| static SplinePtr | createEmpty () |
| Creates an empty spline with all members set to zero. More... | |
| static SplinePtr | createLinear (const vector< Vector3 > &controlPoints, bool looping) |
| Creates a piecewise linear spline with given control points. More... | |
Protected Member Functions | |
| Spline () | |
| Construct spline with member values set to 0. More... | |
| Spline (unsigned int degree, vector< float > coefficientComponents, bool looping) | |
| Construct spline with passed in parameters as member values. More... | |
Spline is a piecewise polynomial curve that defines a path in a 3-dimensional space.
The spline consists of any number of piecewise segments. Each piecewise segment of the spline is a polynomial of the form: y = a + b*x + c*x^2 + d*x^3 + ..., where coefficients a,b,c,d... are Vector3 values that define the spline, x is the time point value from range (0,1) that is used to index the spline, and y is the resulting value of the path.
The degree of a polynomial corresponds with the highest coefficient that is nonzero. For example if c is non-zero but coefficients d and higher are all zero, the polynomial is of degree 2.
The shapes that polynomials can make are as follows: degree 0: Constant, only a is non-zero. Example: y = 3, a constant, uniquely defined by one point. degree 1: Linear, b is highest non-zero coefficient. Example: y = 1 + 2*x. A line, uniquely defined by two points. degree 2: Quadratic, c is highest non-zero coefficient. Example: y = 1 - 2*x + x^2. A parabola, uniquely defined by three points. degree 3: Cubic, d is highest non-zero coefficient. Example: y = -1 - 7/(2*x) + 3/(2*x^3). A cubic curve (which can have an inflection, at x = 0 in this example), uniquely defined by four points.
The degree three polynomial - known as a cubic polynomial - is the one that is most typically chosen for constructing smooth curves in computer graphics. It is used because it is the lowest degree polynomial that can support an inflection - so we can make interesting curves, and it is very well behaved numerically - that means that the curves will usually be smooth.
The spline can be defined to be either looping or open. In a looping spline, the last piecewise segment is connected to the first segment, and the time point can be used in a modular fashion. In an open spline, the time point is clamped between the first and last piecewise segment.
Create a spline from given control points.
Evaluate a spline at a given time point.
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explicitprotected |
Construct spline with member values set to 0.
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explicitprotected |
Construct spline with passed in parameters as member values.
| degree | Degree of the polynomial. |
| coefficientComponents | Piecewise polynomial coefficients, where the highest order coefficient is the first element. |
| looping | Controls whether the spline is looping. |
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static |
Creates an empty spline with all members set to zero.
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static |
Creates a spline from polynomial coefficients.
Compared to the createLinear and createCatmullRom methods, this function is suitable only when you have already precalculated the coefficients.
| degree | Degree of the polynomial. |
| coefficients | Polynomial coefficients for each piecewise segment, where the highest order coefficient of the first segment is the first element. This is an vector of polynomials, where each polynomial is an array of coefficients, where each coefficient is a Vector3-value. The number of items in the vector should be a multiple of (degree+1). |
| looping | Controls whether the spline is looping. |
|
static |
Creates a spline from polynomial coefficients.
Compared to the createLinear and createCatmullRom methods, this function is suitable only when you have already precalculated the coefficients.
| degree | Degree of the polynomial. |
| coefficientComponents | Polynomial coefficients for each piecewise segment, where the highest order coefficient of the first segment is the first element. This is an vector of polynomials, where each polynomial is an array of coefficients, where each coefficient is a (x,y,z)-value. This 3-dimensional array is given as flattened to 1-dimension: coefficients[segment][degree][dimension] -> coefficientComponents[segment * degree * dimension]. The number of items in the vector should be a multiple of (degree+1)*dimension. |
| looping | Controls whether the spline is looping. |
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static |
Creates a piecewise linear spline with given control points.
Use this to create a spline that goes linearly through all the given control points.
| controlPoints | Control points as Vector3s. |
| looping | Controls whether the spline is looping. |
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static |
Creates a Catmull-Rom spline with given control points.
The Catmull-Rom spline is a cubic spline that goes through all the given control points. It is a Hermite spline where tangents are calculated automatically as t_n = (p_(n+1) - p_(n-1)) / 2. If looping is disabled, first and last tangents are calculated simply t_0 = p_1 - p_0 and t_N = p_N - p_N-1.
| controlPoints | Control points as Vector3s. |
| looping | Controls whether the spline is looping. |
|
static |
Creates a Catmull-Rom spline with given control points.
The Catmull-Rom spline is a cubic spline that goes through all the given control points. It is a Hermite spline where tangents are calculated automatically as t_n = (p_(n+1) - p_(n-1)) / 2. If looping is disabled, first and last tangents are calculated simply t_0 = p_1 - p_0 and t_N = p_N - p_N-1.
| controlPointComponents | Control points as a 2-dimensional array that is flattened to 1-dimension: controlPoints[point][dimension] -> controlPointComponents[point * dimension]. The number of items in the vector should be a multiple of dimension. |
| looping | Controls whether the spline is looping. |
| SplinePtr kanzi::Spline::clone | ( | ) | const |
Clone this spline and return the cloned spline.
| unsigned int kanzi::Spline::getSegmentCount | ( | ) | const |
Get the spline segment count.
Use this to find out how many distinct segments a spline has. The spline consists of segments between the points: A looping spline has as many segments as control points, and a non-looping spline has one segment less.
| Vector3 kanzi::Spline::getPoint | ( | float | timePoint | ) | const |
Evaluates the value of the spline in the given time point.
Integer part of time point defines the segment index and decimal part of time point defines the location in that segment. Use getSegmentCount to get the number of segments. In a non-looping spline, the segment index is clamped to valid values. In a looping spline, the valid segment indices are extended with modular arithmetic.
| timePoint | Segment index and position inside the segment. |
| const vector<float>& kanzi::Spline::getCoefficientComponents | ( | ) | const |
Get the polynomial coefficients components for a spline.
| unsigned int kanzi::Spline::getDegree | ( | ) | const |
Get degree of the spline polynomial.
Use this to get what degree of polynomial equation a spline uses.
| bool kanzi::Spline::isLooping | ( | ) | const |
Indicates whether the spline is looping.
