Tag Archives: application

GEOSTRUCT now with Save and Fetch for your constructions

Thanks to intense use of the internet I finally found a simple, understandable way of implementing Save and Fetch operations, enabling the keeping and reusing of any construction.

Here is a reminder of the application (app, program, software, whatever), with the file handling operations:

The user panel and a simple example of three points on a circle, with the bisectors of the pairs of points.

The history panel showing the actions that have been carried out

The Save popup,and, below, the resulting text file.

There is now a not quite finished Spanish option – just click “ESPANOL”

Also a modified “move object” procedure for use with a tablet,or even a smartphone.

The whole application is constructed as a web page, and to run it just click this link: geostruct

Filed under education, geometry app, geostruct, math, math apps, teaching, web based

Halving a triangle, follow-up number two, pursuing the hyperbola

Halving the triangle, any triangle, led to the equation XY = 2 as the condition on the points on two sides of the triangle, distant X and Y from the vertex.
The envelope of this set of lines turned out to be a hyperbola.
But XY = 2 defines a hyperbola – what is the connection ?

I took xy = 1 for the condition, on a standard xy grid, and wrote it as representing a function x —-> y, namely y = 1/x
The two points of interest are then (x,0) on the x-axis and (0,1/x) on the y-axis.
We need the equation of the line joining these two points, so first of all we have to see that our x, above, is telling us which line we are talking about, and so it is a parameter for the line.
We had better give it a different name, say p.
Now we can find the equation of the line in x,y form, using (p,0) and (0,1/p) for the two points:
(y – 0)/(1/p – 0) = (x – p)/(0 – p)
which is easier to read as yp = -x/p + 1, and easier to process as yp2 = -x + p

Now comes the fun bit !
To find the envelope of a set of straight lines we have to find the points of intersection of adjacent lines (really? adjacent?). To do this we have to find the partial derivative (derivative treating almost everything as constant) of the line equation with respect to the parameter p. A later post will reveal all about this mystifying procedure).
So do it and get  2yp = 1

And then eliminate p from the two equations, the line one and the derived one:
From the derived equation we get p = 1/(2y), so substituting in the line equation gives 1 = 2xy
This is the equation of the envelope, and written in functional form it is
y = 1/(2x), or (1/2)(1/x)
Yes ! Another rectangular hyperbola, with the same asymptotes.
(write it as xy = 1/2 if you like)

Now I thought “What will this process do with y = x2 ?”
So off I go, and to cut a long story short I found the following:
For y = x2 the envelope was y = (-1/4)x2, also a multiple of the original, with factor -1/4

Some surprise at this point, so I did it for 1/x2 and for x3
Similar results: Same function, with different factors.
Try it yourself ! ! ! ! ! ! !

This was too much ! No stopping ! Must find the general case ! (y = xk)
Skipping the now familiar details (left to the reader, in time honoured fashion) I found the following:

Original equation: y = xk

Equation of envelope: y = xk multiplied by -(k-1)k-1/kk

which I did think was quite neat.
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The next post will be the last follow-up to the triangle halving.

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Constructions made with GEOSTRUCT, an online browser application:

To get geostruct from the net click
http://www.mathcomesalive.com/geostruct/geostructforbrowser1.html

http://www.mathcomesalive.com/geostruct/geostruct basics.doc

Filed under algebra, calculus, construction, envelope, geometrical

GEOSTRUCT, a program for investigative geometry

I have been developing this computer software / program / application for some years now, and it is now accessible as a web page, to run in your browser.

It provides basic geometric construction facilities, with lines, points and circles, from which endless possibilities follow.

Just try it out, it’s free.

Click on this or copy and paste for later : www.mathcomesalive.com/geostruct/geostructforbrowser1.html

.Here are some of the basic features, and examples of more advanced constructions, almost all based on straightedge and compass, from “make line pass through a point” to “intersection of two circles”, and dynamic constructions with rolling and rotating circles.

Two lines, with points placed on them

Three random lines with two points of intersection generated

Five free points, three generated circles and a center point

Three free points, connected as point pairs, medians generated

Two free circles and three free points, point pairs and centers generated

GIF showing points of intersection of a line with a circle

Construction for locus of hypocycloid

GIF showing a dilation (stretch) in the horizontal direction

Piston and flywheel

Construction for circle touching two circles

Construction for the locus of a parabola, focus-directrix definition.

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Filed under education, geometry, math, operations, teaching