1. Approximation of circular arcs by quartic splines
Ahn, Young Joon
Dept. Math. Edu., Chosun Univ.

2. History approximation of order 2n
authors
target curve
spline curve
year
de Boor, Hollig, Sabin
planar curve
cubic spline
1987
Dokken, Deahlen, Lyche, Morken
circular arc
1990
Goldapp
1991
Ahn, Kim
quartic and quintic spline
1997
Floater
conic
spline of odd degree

3. Circular arc C

4. Quartic spline
Quartic basis of spline
Control coefficients

5. Quartic Bezier curves
coefficients
Bernstein polynomial

6. Hausdorff distance Error function Alternative error function
They have the maximum at the same positions

7. Approximation Method C0 end-point interpolation &
symmetric

8. Theorem[AhnKim]
b and C have geometric contact of order k (GCk) at t0
b and C have GC0 at t=1/2
b and C have GC2 at t=1/2

9. Error Analysis Polynomial of degree eight with zeros at

10. Comparison of Hausdorff distance
Quartic approximation
Zeros of at
dH(b,c)
GC3 end-points
0,0,0,0,1,1,1,1
3.50*10-5
GC2 end-points and GC1 at middle point
0,0,0,1/2,1/2,1,1,1
3.55*10-6
GC1 end-points and GC3 at middle point
0,0,1/2,1/2,1/2,1/2,1,1
2.03*10-6

11. future works Extension this method to all even degree
(2) Extension to conic approximation
(3) Extension to surface case (quadric surface)