Bresenham tasks in ATS: Difference between revisions
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[[File:Bresenham tasks ATS some quadratics.png|alt=An oval made of four quadratic splines, each spline in a different color, with an aqua background.]]
[[File:Bresenham tasks ATS some cubics.png|alt=Cubic splines in multiple colors. There is a variety of cusps, inflections, self-intersections, closedness, and sometimes not the whole spline fits in the window.]]
===A Maxima script I used===
It might be useful if I posted the Maxima script I used to help me devise some of the Bézier curve code. The Maxima commands are not organized well, nor documented well, but nevertheless it might prove useful:
<syntaxhighlight lang="maxima">
/* sbern <--> spow */
T1i : matrix(
[1, 0],
[0, 1]);
T1 : invert(T1i);
T2i : matrix(
[1, 0, 0],
[1, 1, 1],
[0, 0, 1]);
T2 : invert(T2i);
T3i : matrix(
[1, 0, 0, 0],
[2, 1, 0, 1],
[1, 0, 1, 2],
[0, 0, 0, 1]);
T3 : invert(T3i);
expand(T1.matrix([c0],[c1]));
expand(T2.matrix([c0],[c1],[c2]));
expand(T3.matrix([c0],[c1],[c2],[c3]));
/* sbern <--> mono */
S1i : transpose(matrix(
[1, -1],
[0, 1]));
S1 : invert(S1i);
S2i : transpose(matrix(
[1, -2, 1],
[0, 1, -1],
[0, 0, 1]));
S2 : invert(S2i);
S3i : transpose(matrix(
[1, -3, 3, -1],
[0, 1, -2, 1],
[0, 0, 1, -1],
[0, 0, 0, 1]));
S3 : invert(S3i);
F1 : expand(S1i.T1i.matrix([c0],[c1]));
F2 : expand(S2i.T2i.matrix([c0],[c1],[c2]));
F3 : expand(S3i.T3i.matrix([c0],[c1],[c2],[c3]));
/* portion t0..t1 */
g(x) := t0 + ((t1 - t0) * x);
f2(x) := expand(F2.matrix([1],[x],[x**2]));
expand(T2.S2.matrix(
[coeff(f2(g(t)), t, 0)],
[coeff(f2(g(t)), t, 1)],
[coeff(f2(g(t)), t, 2)]));
f3(x) := expand(F3.matrix([1],[x],[x**2],[x**3]));
Solution3 : expand(T3.S3.matrix(
[coeff(f3(g(t)), t, 0)],
[coeff(f3(g(t)), t, 1)],
[coeff(f3(g(t)), t, 2)],
[coeff(f3(g(t)), t, 3)]));
Solution3[1];
Solution3[2];
Solution3[3];
Solution3[4];
/* derivatives and critical points */
/* In s-power format: */
Mx : S3i.T3i.matrix([cx0],[cx1],[cx2],[cx3]);
My : S3i.T3i.matrix([cy0],[cy1],[cy2],[cy3]);
fx(t) := expand(transpose(Mx).matrix([1],[t],[t**2],[t**3]));
fy(t) := expand(transpose(My).matrix([1],[t],[t**2],[t**3]));
fxp(t) := diff(fx(t), t);
fxpp(t) := diff(fxp(t), t);
fyp(t) := diff(fy(t), t);
fypp(t) := diff(fyp(t), t);
cross(t) := expand((fxp(t) * fypp(t)) - (fyp(t) * fxpp(t)));
expand(fxp(t));
expand(fyp(t));
expand(cross(t) / 2); /* Notice that this is quadratic. */
/* Translating the first endpoint to (0,0), and the second to
cx3 - cx0. */
Mx : S3i.T3i.matrix([0],[cx1],[cx2],[cx3m0]);
My : S3i.T3i.matrix([0],[cy1],[cy2],[cy3m0]);
fx(t) := expand(transpose(Mx).matrix([1],[t],[t**2],[t**3]));
fy(t) := expand(transpose(My).matrix([1],[t],[t**2],[t**3]));
fxp(t) := diff(fx(t), t);
fxpp(t) := diff(fxp(t), t);
fyp(t) := diff(fy(t), t);
fypp(t) := diff(fyp(t), t);
cross(t) := expand((fxp(t) * fypp(t)) - (fyp(t) * fxpp(t)));
expand(fxp(t));
expand(fyp(t));
expand(cross(t) / 2); /* Notice that this is quadratic. */
</syntaxhighlight>
Some of the calculations are done by assigning names to the coefficients of splines in the s-power basis, finding the corresponding coefficients in the ordinary monomial basis, then (using Maxima's inherent capabilities) performing an operation such as composition or differentiation. It is a sloppy script.
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