# Tag Archives: translation

## Rigid motions are actually useful !

I am currently reconstructing my geometrical construction application, Geostruct, to run in a web page using javascript.

One of the actions is to find the points of intersection of a straight line with a circle. Here is a gif showing the result:

The algebra needed to solve the two simultaneous equations is straightforward, but a pain in the butt to get right and code up, so I thought “Why not solve the equations for the very simple case of the circle centered at the origin and the line vertical, at the same distance (a) from the centre of the circle

Then it is a simple matter of  rotating the two points (a,b) and (a,-b) about the origin, through the angle made by the original line to the vertical, and then translating the circle back to its original position, the translated points are then the desired points of intersection.

The same routine can be used for the intersection of two circles, with a little bit of prior calculation.

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Filed under geometry, math

## Rigid Transformations – Coordinate axes

A simple diagram with original axes in blue.
The coordinates of point E are (1,1)
A translation defined by x -> x + 2, y -> y + 1 moves point E to point D, with coordinates (3,2)

If the x axis is moved 2 steps left and the y axis is moved one step down then the coordinates of the original point E in the moved axes are (3,2)

This will be the case for any original point – the coordinates of each one of them will be the same as the coordinates of their new positions under the translation (in the original coordinate system).

This can be seen to be true for the other rigid motions, for example
rotation about the origin through an angle theta is equivalent to a rotation about the origin of the axes through an angle minus theta. So there is a one to one correspondence between rigid motions and change of axes (scales preserved).

On a lighter note, it does seem easier to rotate a pair of axes than rotating the whole plane ! ! !

Filed under algebra, education, geometry, language in math

## Congruence. Transformations. Definitions. Unnecessary.

Once again I take the Common Core to task.

This time their obsession with transformations.

There is nothing wrong with learning about transformations but it is silly to attempt a definition of congruence in terms of transformations, as we will see. Rigid transformations of the plane (translations and rotations) preserve lengths and angles, and so a rigid transformation of a figure gives a new figure which is “the same”. Also, and ignored by CCSS is the vital fact that such a transformation can take an image to ANY position in the plane, where position is identified as “point + direction”.

So this is what is written:

High school geometry: Congruence
Preamble
The concepts of congruence, similarity, and symmetry can be understood from the perspective of geometric transformation. Fundamental are the rigid motions: translations, rotations, reflections, and combinations of these, all of which are here
assumed to preserve distance and angles (and therefore shapes generally).
In the approach taken here, two geometric figures are defined to be congruent if there is a sequence of rigid motions that carries one onto the other. This is the principle of superposition. (me – This last is garbage – see at end)

So here goes, an attempt to see if line segment FQ is congruent to BA

We slide BA in the direction B to F until B coincides with F – this is our translation.  A is now at the point K

We then rotate BA around point F until it is on top of  FQ, it is now in position FL. This is all possible as the position of the (simple) figure FQ must be known.

Have we carried BA onto FQ ? Not necessarily. It depends on where point Q is. The only way that L and Q can be the same point is a) if we know the length of segment FQ and b) this length is equal to the length of FL, which is the length of BA.

Therefore the only way that BA can be moved and placed on FQ is if they are the same length.

What have we done?  We have shown that if BA and FQ (or any two line segments) have the same length then there is a rigid transformation carrying one to the other. Hence by the definition of “congruent” BA and FQ are congruent.

In simple words, If BA and FQ have the same length then they are congruent, so simplicity (a highly desirable aim in math) is better served by the simpler definition:

TWO LINE SEGMENTS ARE DEFINED TO BE CONGRUENT IF THEY HAVE THE SAME LENGTH (AND not OTHERWISE)

The dragging in of transformations really muddies the water.
Here is part of the High School geometry detail

Understand congruence in terms of rigid motions
6. …..; given two figures, use the definition of congruence in terms of rigid motions
to decide if they are congruent.

and here is some dictionary stuff on superposition

http://www.thefreedictionary.com/superposition
2.  The principle by which the description of the state of a physical system can be broken down into descriptions that are themselves possible states of the system. For example, harmonic motion, as of a violin string, can be analyzed as the sum of harmonic frequencies or harmonics, each of which is itself a kind of harmonic motion; harmonic motion is therefore a superposition of individual harmonics.
3. superposition – (geometry) the placement of one object ideally in the position of another one in order to show that the two coincide
locating, positioning, emplacement, location, placement, position – the act of putting something in a certain place
4. superposition – the placement of one thing on top of another
locating, positioning, emplacement, location, placement, position – the act of putting something in a certain place

There is no Principle of Superposition in geometry!