Centripetal Catmull–Rom spline
In computer graphics, centripetal Catmull–Rom spline is a variant form of Catmull-Rom spline [1] formulated by Edwin Catmull and Raphael Rom according to the work of Barry and Goldman.[2] It is a type of interpolating spline (a curve that goes through its control points) defined by four control points , with the curve drawn only from to .
Definition
Let denote a point. For a curve segment defined by points and knot sequence , the centripetal Catmull-Rom spline can be produced by:
where
and
in which ranges from 0 to 1 for knot parameterization, and with . For centripetal Catmull-Rom spline, the value of is . When , the resulting curve is the standard Catmull-Rom spline (uniform Catmull-Rom spline); when , the product is a chordal Catmull-Rom spline.
Plugging into the spline equations and shows that the value of the spline curve at is . Similarly, substituting into the spline equations shows that at . This is true independent of the value of since the equation for is not needed to calculate the value of at points and .
Advantages
Centripetal Catmull–Rom spline has several desirable mathematical properties compared to the original and the other types of Catmull-Rom formulation.[3] First, it will not form loop or self-intersection within a curve segment. Second, cusp will never occur within a curve segment. Third, it follows the control points more tightly.
Other uses
In computer vision, centripetal Catmull-Rom spline has been used to formulate an active model for segmentation. The method is termed active spline model.[4] The model is devised on the basis of active shape model, but uses centripetal Catmull-Rom spline to join two successive points (active shape model uses simple straight line), so that the total number of points necessary to depict a shape is less. The use of centripetal Catmull-Rom spline makes the training of a shape model much simpler, and it enables a better way to edit a contour after segmentation.
Code example
The following is an implementation of the Catmull–Rom spline in Python.
import numpy
import pylab as plt
def CatmullRomSpline(P0, P1, P2, P3, nPoints=100):
"""
P0, P1, P2, and P3 should be (x,y) point pairs that define the Catmull-Rom spline.
nPoints is the number of points to include in this curve segment.
"""
# Convert the points to numpy so that we can do array multiplication
P0, P1, P2, P3 = map(numpy.array, [P0, P1, P2, P3])
# Calculate t0 to t4
alpha = 0.5
def tj(ti, Pi, Pj):
xi, yi = Pi
xj, yj = Pj
return ( ( (xj-xi)**2 + (yj-yi)**2 )**0.5 )**alpha + ti
t0 = 0
t1 = tj(t0, P0, P1)
t2 = tj(t1, P1, P2)
t3 = tj(t2, P2, P3)
# Only calculate points between P1 and P2
t = numpy.linspace(t1,t2,nPoints)
# Reshape so that we can multiply by the points P0 to P3
# and get a point for each value of t.
t = t.reshape(len(t),1)
A1 = (t1-t)/(t1-t0)*P0 + (t-t0)/(t1-t0)*P1
A2 = (t2-t)/(t2-t1)*P1 + (t-t1)/(t2-t1)*P2
A3 = (t3-t)/(t3-t2)*P2 + (t-t2)/(t3-t2)*P3
B1 = (t2-t)/(t2-t0)*A1 + (t-t0)/(t2-t0)*A2
B2 = (t3-t)/(t3-t1)*A2 + (t-t1)/(t3-t1)*A3
C = (t2-t)/(t2-t1)*B1 + (t-t1)/(t2-t1)*B2
return C
def CatmullRomChain(P):
"""
Calculate Catmull Rom for a chain of points and return the combined curve.
"""
sz = len(P)
# The curve C will contain an array of (x,y) points.
C = []
for i in range(sz-3):
c = CatmullRomSpline(P[i], P[i+1], P[i+2], P[i+3])
C.extend(c)
return C
# Define a set of points for curve to go through
Points = [[0,1.5],[2,2],[3,1],[4,0.5],[5,1],[6,2],[7,3]]
# Calculate the Catmull-Rom splines through the points
c = CatmullRomChain(Points)
# Convert the Catmull-Rom curve points into x and y arrays and plot
x,y = zip(*c)
plt.plot(x,y)
# Plot the control points
px, py = zip(*Points)
plt.plot(px,py,'or')
plt.show()
UNITY C# IMPLEMENTATION
using UnityEngine;
using System.Collections;
using System.Collections.Generic;
public class Catmul : MonoBehaviour {
//Use GameObject in 3d space as your points or define array with desired points
public GameObject[] points;
//Store points on the Catmull curve so we can visualize them
List<Vector2> newPoints = new List<Vector2>();
//How many points you want on the curve
float amountOfPoints = 10.0f;
//set from 0-1
public float alpha = 0.5f;
/////////////////////////////
void Update()
{
CatmulRom();
}
void CatmulRom()
{
newPoints.Clear();
Vector2 p0 = new Vector2(points[0].transform.position.x, points[0].transform.position.y);
Vector2 p1 = new Vector2(points[1].transform.position.x, points[1].transform.position.y);
Vector2 p2 = new Vector2(points[2].transform.position.x, points[2].transform.position.y);
Vector2 p3 = new Vector2(points[3].transform.position.x, points[3].transform.position.y);
float t0 = 0.0f;
float t1 = GetT(t0, p0, p1);
float t2 = GetT(t1, p1, p2);
float t3 = GetT(t2, p2, p3);
for(float t=t1; t<t2; t+=((t2-t1)/amountOfPoints))
{
Vector2 A1 = (t1-t)/(t1-t0)*p0 + (t-t0)/(t1-t0)*p1;
Vector2 A2 = (t2-t)/(t2-t1)*p1 + (t-t1)/(t2-t1)*p2;
Vector2 A3 = (t3-t)/(t3-t2)*p2 + (t-t2)/(t3-t2)*p3;
Vector2 B1 = (t2-t)/(t2-t0)*A1 + (t-t0)/(t2-t0)*A2;
Vector2 B2 = (t3-t)/(t3-t1)*A2 + (t-t1)/(t3-t1)*A3;
Vector2 C = (t2-t)/(t2-t1)*B1 + (t-t1)/(t2-t1)*B2;
newPoints.Add(C);
}
}
float GetT(float t, Vector2 p0, Vector2 p1)
{
float a = Mathf.Pow((p1.x-p0.x), 2.0f) + Mathf.Pow((p1.y-p0.y), 2.0f);
float b = Mathf.Pow(a, 0.5f);
float c = Mathf.Pow(b, alpha);
return (c + t);
}
//Visualize the points
void OnDrawGizmos()
{
Gizmos.color = Color.red;
foreach(Vector2 temp in newPoints2)
{
Vector3 pos = new Vector3(temp.x, temp.y, 0);
Gizmos.DrawSphere(pos, 0.3f);
}
}
}
Note: If you need to implement it in 3d space with Vector3 points, just extend the float a in function GetT to this : Mathf.Pow((p1.x-p0.x), 2.0f) + Mathf.Pow((p1.y-p0.y), 2.0f) + Mathf.Pow((p1.z-p0.z), 2.0f); and convert all your Vectors2 to Vectors3.
See also
References
- ↑ E. Catmull and R. Rom. A class of local interpolating splines. Computer Aided Geometric Design, pages 317-326, 1974.
- ↑ P. J. Barry and R. N. Goldman. A recursive evaluation algorithm for a class of Catmull–Rom splines. SIGGRAPH Computer Graphics, 22(4):199-204, 1988.
- ↑ Yuksel, C.; Schaefer, S.; Keyser, J. (2011). "Parameterization and applications of Catmull-Rom curves". Computer-Aided Design. 43: 747–755.
- ↑ Jen Hong, Tan; U. R., Acharya (2014). "Active spline model: A shape based model—interactive segmentation". Digital Signal Processing. 35: 64–74.
External links
- Implementation in Java
- Simplified implementation in C++
- Interactive generation via Python, in a Jupyter Notebook