When the first version of Sketchpad was released, you didn't see many
constructions involving curves other than circles. With Version 2, people did a
lot of curve stitching and curved locus tracing. With Version 3, you have arcs,
and you have dynamic loci. That, combined with dynamic plotting, makes for a lot
of wonderful curve construction!
The sketches on this page are available individually (below) or can be downloaded
as a package.
Squaring the Circle
This is a quick demo of what it means to square the circle that Dan Bennett
at Key Curriculum Press did using a very early alpha of Sketchpad/3. After years
of explaining that Sketchpad was restricted to Euclidean constructions, he
couldn't resist. The buttons animate a circle deforming into a square and vice
versa brought to you by arcs and by dynamic transformations tied to calculated
This sketch rolls a Reuleaux triangle between two "plates," allowing you to
change the size and shape of the triangle and to trace the path of the center of
the triangle as it rolls. It makes use of arcs to handle the rounded vertices of
The search for the ideal conics sketch continues. Scott Steketee, programmer
for the Windows version of Sketchpad created this one. It's based on the
definition of a conic as the locus of points which are equally distant from a
given point F1 (the focus) and a given circle (the directrix).
In this construction, the focus F1 and point P on the circle are kept fixed.
The radius of the circle is changed by moving the center F2 along the line
joining F1 and P.
You can adjust the size of the circle, and change the eccentricity of the
conic, by moving a point labeled "drag me," (not shown) or you can double-click
an "Animate" button.
This curve demonstrates Bezier's description of a cubic. Four control points
define the curve: two "endpoints" on the curve, and two which specify tangent
velocities of the curve at the two endpoints. The sketch also demonstrates
connecting Bezier curves by joining endpoints, and smoothly connecting
Bezier curves by both joining endpoints and matching tangents. Bezier curves
frequently appear in computer-graphics, computer-modeling, and
The sketch was constructed by entering the curve's parametric equation (into
the Calculator) based on some variable t (the parameter) and the coordinates of
the four control points. Two separate equations were used to calculate the X and
Y components of a point on the curve for some value of the parameter. These two
equations were then plotted as a single point in the sketch, and the curve was
then defined as the locus of that point as t varied from 0 to 1.
Bezier Patch (Warning: 1136K)
This "bi-parametric cubic patch" is the result of multiplying two
independent Bezier curves, resulting in a surface. Again, control points (16 of
them now) determine the curvature of the surface, both by "on-patch" points and
by "tangent velocity" points. Bicubic patches are a standard representation of
non-planar surfaces in computer modeling situations. Warning: You will
need a very fast computer to manipulate this sketch--and even then it will
be slow! (Each time Sketchpad draws the patch, it must evaluate a cubic function
almost 64,000 times. In real modeling situations, more efficient algorithms are
used for drawing bicubic patches.) On a Macintosh, you'll also want to allocate
extra memory to the Sketchpad application: we recommend as large a partition as
Return to the Foyer.
Sketches, scripts, and web pages by Bill Finzer
and Nick Jackiw.