QUICKLY CREATE ACCURATE DRAWINGS FOR TEST and PRESENTATIONS
Introducing Math Illustrations, an easy intuitive way to create geometric diagrams for documents and presentations.
By combining the constraint-based architecture of Geometry Expressions with easy-to-use drawing and graphing features, Math Illustrations lets you create more effective geometry figures for your students in less time.
Got Clock Apps? If you need some time, we're sending our geometrical Clock Apps made with Geometry Expressions to keep you entertained and up to the minute! Just click the blue image captions to try out the apps, and you can download them to put on your personal webpage. (These Clocks, with their explanatory apps are not only free of charge, but also advert free!) If you own a copy of Geometry Expressions, you can also download the .gx file to see how they are made and modify them to your preference.
To make your own personalized Clocks or other apps, get your copy of Geometry Expressions.
Clocks Volume 01 The Morphing Clock
Is this clock a circle, or a square, or something in-between, or something beyond?
Not only do the hands move in this clock, but its shape changes with time. At the hour it is circular, at the quarter hours it is square and at the half hour, it has gone beyond square to a concave curved square shape. The morphing is achieved by taking a linear combination of the parametric representation of a square and a circle.
Geometrically, the morph curve is constructed
by joining two corresponding
points on the original curves by a line, and
then positioning a third point at a specific
proportion along this line (Figure 1.1).
In the figure, let parameter t specify the location of the points on the original curves (the red dots on the circle and the square), and parameter s specify the location of the morph curve's deﬁning point (the middle red dot) on the line joining the two original points.
Algebraically, if a(t) is the location of the point on the circle, and b(t) is the location of the point on the square, then the morph curve is deﬁned by:
c(t) = s.b(t)+(1-s).a(t)
If s="0," then c(t)=a(t) and the morph curve is the circle. If s="1," then c(t)=b(t) and the
morph curve is a square. If s is between 0 and 1, the morph curve is an intermediate
If s goes beyond 1, the morph curve is still defined, but now we are in a sense subtracting
the circle from the square and instead of blunting the square's edges, we are exaggerating them.
The form of the morph curve depends on how the original curves are parameterized. Changing the parameterization of the original curves does not change their appearance when displayed, but it does change the appearance of the morph curve. Figure 1.2 illustrates this.
A similar morphing operation can be defined between any pair of parametric curves, open or closed. In Figure 1.3, this is illustrated by morphing a circle into a line segment. Notice that the need to open up the circle clearly illustrates the start and end points of its parametrization. Moving the location of the destination line segment relative to the start of the circle's parametrization yields different morph curves. For example, try dragging the ends of the line segment so it is positioned to the right rather than to the left of the circle.
Another experiment to try is to drag the
end points of the line so that it is upside down.