As like hermit spline cardinal splines also interpolating piecewise cubics with specified endpoint tangents at the boundary of each section.
But in this spline, we need not have to input the values of endpoint tangents.
In cardinal spline values of slope at control point calculated from two immediate neighbor control points.
Its spline section is completely specified by the 4-control points.
Fig from Cardinal Splines, parametric point function p(u) for a cardinal spline section between control points pk and pk+1.
The middle two endpoints of curve section and other two used to calculate the slope of endpoints.
Now parametric equation for cardinal spline is:
Where parameter t called tension parameter since it controls how loosely or tightly the cardinal spline fit the control points.
Fig. Effect of the tension parameter on the shape of a cardinal spline section
When t = 0 this class of curve is referred to as Catmull-rom spline or overhauser splines.
Using the similar method like hermit we can obtain:
Where polynomial 𝐶𝐴𝑅(𝑢) 𝑓𝑜𝑟 𝑘 = 0,1,2,3 are the cardinals blending functions.
It is extension of cardinal spline
Moreover, Two additional parameters introduced into the constraint equation for defining Kochanek-Bartels spline to provide more flexibility in adjusting the shape of curve section.
Also, For this parametric equations are as follows:
Where ‘t’ is tension parameter same as used in the cardinal spline.
B is biased parameter and C the continuity parameter.
In this Cardinal Splines, parametric derivatives may not continuous across section boundaries.
Also, Bias B is used to adjust the amount that the curve bends at each end of a section.
- Fig. Effect of bias parameter on the shape of a Kochanek-Bartels spline section.
- Moreover, Parameter c used to controls continuity of the tangent vectors across the boundaries of a section. If C nonzero there is the discontinuity in the slope of the curve across section boundaries.
- It used in animation paths in particular abrupt change in motion which simulated with nonzero values for parameter C.