Analytically we can define curve in 2D :

*implicitly*- F(x,y) = 0*explicitly*- y = f(x)*parametric*- x = x(t); y = y(t)

**Example: **Let's have a circle with centre in dot S[0,0] and radius
r.

*Implicitly*(**x**^{2}) + (y^{2}) = r^{2}-
*Explicitly***y = sqrt(r**^{2}-x^{2}) *Parametric***x = cos t; y = sin t**for**t ∈<0,2p>**

Applet Select one point on circle.

Construction of an interpolative curve is a construction
of a curve that crosses given nodal dots. In numerical mathematics
interpolations through the medium of polynoms are used to calculate a certain
integral. In computer graphics, interpolation is used to draw a curve for
given instruction dots crossed by theinterpolative curve .
Problem is how to define parameter value, for which
we will obtain given operative (determining) dot of interpolative curve.
The most known interpolative curve is **Ferguson's cubic**. The
name **cubic** is used for curves of the 3rd grade.

This curve was established in year 1964 J.C. Ferguson. It is determined by two operative dots V1 and V2 and values of vectors in these dots v1 and v2. Curve starts in dot V1 and ends in dot V2. The form of curve is determined by direction and size of vectors v1 and v2. If both vectors are zero, then resultant curve will be line. Ferguson's cubic can be written as

| 2 -2 1 1 | | V1 | B(t)= T . | -3 3 -2 -1 |.| V2 |, | 0 0 1 0 | | v1 | | 1 0 0 0 | | v2 |where

**F**_{1}(t) = 2t^{3}- 3t^{2}+1**F**_{2}(t) = 2t^{3}+ 3t^{2}**F**_{3}(t) = t^{3}- 2t^{2}+t**F**_{4}(t) = t^{3}- t^{2}.

Ferguson's cubic

Source code

For aproximative curve we have given some dots, but mostly it need not cross them. Grounds of construction of such curves were build by Bezier and Casteljau in years 1959-62 and it lies in approximation of the given curve by means of operative polygon (broken line), which determines the resultant form of the curve.

Parabola is obviously given by two of its dots and by contact dotsdots in them.

For a line AC, where dot B ∈ AC is defined ratio (ABC) = (B-A)/(C-B).

Let's assume that we have got given dots A, D and C and t ∈<0,1>. For dots E, B and F then it is valid : ratio (AED) = ratio (EBF) = ratio (DFC). Construction of a parabola lies in drawing a dot of parabola B with parameter r = t /(1-t).

Then 1. computes the point E, F : E : ratio(AED) = r ( <=> E = (1-t)*A + t*D ) F : ratio(DFC) = r ( <=> F = (1-t)*D + t*C ) 2. compute the point B : B : ratio(EBF) = r ( <=> B = (1-t)*E + t*F ) 3. The point B is the parabola with the corresponding parameter t

Source code

Source code

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