This Brief Investigates The Intersections That Occur Between Three Different Areas Of Study That Normally Would Not Touch Each Other: Odf, Spline Theory, And Topology.the Least Squares Orthogonal Distance Fitting (odf) Method Has Become The Standard Technique Used To Develop Mathematical Models Of The Physical Shapes Of Objects, Due To The Fact That It Produces A Fitted Result That Is Invariant With Respect To The Size And Orientation Of The Object. It Is Normally Used To Produce A Single Optimum Fit To A Specific Object; This Work Focuses Instead On The Issue Of Whether The Fit Responds Continuously As The Shape Of The Object Changes. The Theory Of Splines Develops User-friendly Ways Of Manipulating Six Different Splines To Fit The Shape Of A Simple Family Of Epitrochoid Curves: Two Types Of Bzier Curve, Two Uniform B-splines, And Two Beta-splines.