# Hermit Interpolation

Hermit Interpolation is the important topic of the Computer Graphics. Moreover, It is the Important subject of the Computer Science & Technological field.

It is named after French mathematician Charles hermit

Moreover, It is an interpolating piecewise cubic polynomial with the specified tangent at each control points.

It adjusted locally because each curve section depends on its endpoints only.

Parametric cubic point function for any curve section then given by:

(0) = đ_{đ}

(1) = đ_{đ+1}

đ^{â˛}(0) = đđ_{đ}

đ^{â˛}*â˛*(1) = đđ_{đ+1}

##### Where dp_{k} & dp_{k+1} are values of parametric derivatives at point p_{k} & p_{k+1} respectively. //Â Hermit Interpolation //

Vector equation of cubic spline is:

(đ˘) = đđ˘^{3 }+ đđ˘^{2 }+ đđ˘ + đ

Where x component of p is

đĽ(đ˘) = đ_{đĽ}đ˘^{3 }+ đ_{đĽ}đ˘^{2 }+ đ_{đĽ}đ˘ + đ_{đĽ} and similarly y & z components

Matrix form of above equation is

##### Now substitute endpoint value of u as 0 & 1 in above equation & combine all four parametric equations in matrix form: //Â Hermit Interpolation //

Where đť(u) for k=0, 1, 2, 3 referred to as blending functions because of that blend the boundary constraint values for curve section.

##### The shape of the four hermit blending function given below.

Fig. the hermit blending functions.

Moreover, Hermit Interpolation curves are used in digitizing application where we input the approximate curve slope means DP_{kÂ }&Â DP_{k+1}.

But in an application where this input difficult to approximate at that place, we cannot use hermit curve.

**Related Terms**

Computer Graphics, Spline Representations, Cubic Spline Interpolation Methods, Polygon Meshes

## Leave a Reply