[alogo] Conic-Arc defined through a rational Bezier curve

Curve of degree 2, determined by 3 control points A, B, C, using the Bernstein polynomials Bn,i(x) and 3 additional (positive) numbers w1, w2, w3, called weights. The curve is defined by the equation:
c(t) = (1/BS(x))*(B2,0(x)*w1*A +B2,1(x)*w2*B + B2,2(x)*w3*C),
where the function BS(x) = B2,0(x)*w1+B2,1(x)*w2 +B2,2(x)*w3.
Here the curve is constructed using a small script (EUC_Scripts\EUC_Curves\ConicThroughRationalBezier.txt). The parameter t varies in the interval [0,1].
The points D1, D2, D3 on the Bezier arc are auxiliary points which in addition to A and C define five points through which we pass a conic. Every conic can be described in this way. Modifying the control points A, B, C does not change the kind of the conic, whereas changing the weights wi we obtain various kinds of conics. The integer nPts gives the number of interpolation points used to draw the Bezier arc.

[0_0] [0_1] [0_2]
[1_0] [1_1] [1_2]

See Also



Duncan Marsh. Applied Geometry for Computer Graphics and CAD Berlin, Springer, 1999, p. 135

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