# Cubic Spline Interpolation Methods

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

- The Cubic Spline Interpolation Methods are mostly used for representing the path of moving object or existing object shape or drawing.
- Sometimes it also used to design the object shapes.
- Cubic spline gives reasonable computation on as compared to higher order spline. And more stable compared to lower order polynomial spline. So it is often used for modeling curve shape.

- Fig. A piecewise continuous Cubic Spline Interpolation Methods of n+1 control points. đ
_{đ }= (đĽ_{đ},_{đ},đ§_{đ})Â Where, k=0, 1, 2, 3 …, n - Cubic interpolation splines obtained by fitting the input points with the piecewise cubic polynomial curve that passes through every control point.
- Parametric cubic polynomial for this curve given by

(đ˘) = đ_{đĽ}đ˘^{3 }+ đ_{đĽ}đ˘^{2 }+ đ_{đĽ}đ˘ + đ_{đĽ}

Moreover, (đ˘) = đ_{đŚ}đ˘^{3 }+ đ_{đŚ}đ˘^{2 }+ đ_{đŚ}đ˘ + đ_{đŚ}

(đ˘) = đ_{đ§}đ˘^{3 }+ đ_{đ§}đ˘^{2 }+ đ_{đ§}đ˘ + đ_{đ§} đ¤âđđđ( 0 â¤ đ˘ â¤ 1)

- For above equation, we need to determine for constant a, b, c, and d the polynomial representation for each of n curve section.
- This obtained by settling proper boundary condition at the joints.
- Now we will see the common method for settling this condition.

**Natural Cubic Splines [ Cubic Spline Interpolation Methods ]**

- A natural cubic splines a mathematical representation of the original drafting spline.
- We consider that curve is in đ
^{2}continuity means first and second parametric derivatives of adjacent curve section are same as a control point. - For the âân+1ââ control point, we haven curve section and 4n polynomial constants to find.
- For all interior control points, we have four boundary conditions. The two curve section on either side of control point must have same first & second order derivative at the control points and each curve passes through that control points.
- We get other two conditions as đ
_{0}(first control points) starting & đ(last control point) is an endpoint of the curve. - We still required two conditions for obtaining coefficient values.
- One approach is to set up the second derivative at đ
_{0}& đ_{đ}to be 0. Another approach is to add one extra dummy point at each end. I.e. we add đ_{â1}& đ_{đ+1}then all original control points are interior and we get 4n boundary condition. - Although it is a mathematical model it has a major disadvantage is the change in the control point entire curve changed.
- So it not allowed for local control and we cannot modify part of the curve.

**Related Terms**

Computer Graphics, Spline Representations, Plane Equations, Polygon Meshes

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