<Text-field style="Heading 1" layout="Heading 1">Introduction</Text-field>

An important family of parametric curves is the Bezier Curves, named after the French mathematician Pierre Bezier. They are used in computer-aided design and typography. A cubic Bezier curve is determined by four control points, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2MFEiPUYnRj4vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkkvJSlzdHJldGNoeUdGSS8lKnN5bW1ldHJpY0dGSS8lKGxhcmdlb3BHRkkvJS5tb3ZhYmxlbGltaXRzR0ZJLyUnYWNjZW50R0ZJLyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYnLyUncnNwYWNlR0Zlbi8lKG1pbnNpemVHUSIxRicvJShtYXhzaXplR1EpaW5maW5pdHlGJw==(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), 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(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),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(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) and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiM0YnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2MFEiPUYnRj4vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkkvJSlzdHJldGNoeUdGSS8lKnN5bW1ldHJpY0dGSS8lKGxhcmdlb3BHRkkvJS5tb3ZhYmxlbGltaXRzR0ZJLyUnYWNjZW50R0ZJLyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYnLyUncnNwYWNlR0Zlbi8lKG1pbnNpemVHUSIxRicvJShtYXhzaXplR1EpaW5maW5pdHlGJw==(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), and is defined by the parametric equations for 0\342\211\244t\342\211\2441,

x = f(t) =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

y = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY8LUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2MFEifkYnRj0vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUlZm9ybUdRIUYnLyUnbHNwYWNlR1EkMGVtRicvJSdyc3BhY2VHRlcvJShtaW5zaXplR1EiMUYnLyUobWF4c2l6ZUdRKWluZmluaXR5RictRkA2MFEiPUYnRj1GQ0ZGRkhGSkZMRk5GUC9GU1EmaW5maXhGJy9GVlEvdGhpY2ttYXRoc3BhY2VGJy9GWUZgb0ZaRmduRj8tSSVtc3ViR0YkNiUtRiw2JVEieUYnRi9GMi1GIzYjLUkjbW5HRiQ2JFEiMEYnRj0vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JJW1zdXBHRiQ2JS1GNjYkLUYjNiUtRltwNiRGZm5GPS1GQDYwUSomdW1pbnVzMDtGJ0Y9RkNGRkZIRkpGTEZORlBGXW8vRlZRMG1lZGl1bW1hdGhzcGFjZUYnL0ZZRl5xRlpGZ25GOkY9LUYjNiMtRltwNiRRIjNGJ0Y9LyUxc3VwZXJzY3JpcHRzaGlmdEdGYHBGPy1GQDYwUSIrRidGPUZDRkZGSEZKRkxGTkZQRl1vRl1xRl9xRlpGZ25GP0ZicS1GY282JUZlby1GIzYjRmhwRl5wRjotRmJwNiVGZHAtRiM2Iy1GW3A2JFEiMkYnRj1GZXFGP0ZncUY/RmJxLUZjbzYlRmVvRmByRl5wLUZicDYlRjpGYHJGZXFGZHBGP0ZncUY/LUZjbzYlRmVvRmBxRl5wLUZicDYlRjpGYHFGZXE=

We will give an example of the Bezier cuve with control points 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

(x0, y0) := (4,1); (x1, y1):=(28, 48); (x2,y2):=(50,42); (x3,y3):=(40,5);

f := x0*(1-t)^3 + 3*x1*t*(1-t)^2 + 3*x2*t^2*(1-t) + x3*t^3;

g := y0*(1-t)^3 + 3*y1*t*(1-t)^2 + 3*y2*t^2*(1-t) + y3*t^3;

The plot command in Maple accepts parametrically defined curves using a special list syntax.

plot([f, g, t=0..1], scaling=constrained);

When t=0, we have the starting point (x,y)=(x0, y0),

eval([f,g], t=0);

and when t=1, we have the end point (x,y)=(x3, y3).

eval([f,g], t=1);

However, the other two control points don't lie on the curve, as we can see by adding them to the plot.

plot1 := plot([f, g, t=0..1], color=red):

plot2 := pointplot([[x0, y0], [x1, y1], [x2, y2], [x3, y3]]):

plot3 := plot([[x0, y0], [x1, y1], [x2, y2], [x3, y3]], color=blue):

with(plots):

display([plot1,plot2, plot3],scaling=constrained);

The interior control points do influence the shape of the Bezier curve. For instance, we can prove that the tangent line at 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

Here is the slope of the tangent of the Bezier curve:

slope := diff(g,t)/diff(f,t);

We have seen that at 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The tangent line of the Bezier curve at 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satisfies

tanline1 := y = eval(slope, t=0)*(x-x0) + y0; eval(tanline1, x=28);

and at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiM0YnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn we have

tanline2 := y = eval(slope, t=1)*(x-x3) + y3; eval(tanline2, x=50);

Therefore 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 is on the line tangent at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn, and 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 is on the line tangent at 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

Once a curve is defined, is easy to do some transformations. For example:

Reflect the curve across the x-axis. We can leave LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== alone and change LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2MFEqJnVtaW51czA7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJWZvcm1HUSZpbmZpeEYnLyUnbHNwYWNlR1EwbWVkaXVtbWF0aHNwYWNlRicvJSdyc3BhY2VHRkYvJShtaW5zaXplR1EiMUYnLyUobWF4c2l6ZUdRKWluZmluaXR5RictSSNtaUdGJDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0Yn.

plot([f, -g, t=0..1],scaling=constrained);

Rotate the curve \316\270 degrees clockwise. We can use the roation 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

For example, if \316\270= 60 degrees, then

plot([1/2*f+sqrt(3)/2*g, -sqrt(3)/2*f+1/2*g, t=0..1], scaling=constrained);