Tag Archives: transformation

Linear transformations, geometrically


Following a recent blog post relating a transformation of points on a line to points on another line to the graph of the equation relating the input and output I thought it would be interesting to explore the linear and affine mappings of a plane to itself from a geometrical construction perspective.

It was ! (To me anyway)

These linear mappings  (rigid and not so rigid motions) are usually  approached in descriptive and manipulative  ways, but always very specifically. I wanted to go directly from the transformation as equations directly to the transformation as geometry.

Taking an example, (x,y) maps to (X,Y) with the linear equations

X = x + y + 1 and Y = -0.5x +y

it was necessary to construct a point on the x axis with the value of X, and likewise a point on the y axis with the value of Y. The transformed (x,y) is then the point (X,Y) on the plane.

The construction below shows the points and lines needed to establish the point(X,0), which is G in the picture, starting with the point D as the (x,y)


transform of x

The corresponding construction was done for Y, and the resulting point (X,Y) is point J. Point D was then forced to lie on a line, the sloping blue line, and as it is moved along the line the transformed point J moves on another line

gif for lin affine trans1

Now the (x,y) point (B in the picture below, don’t ask why!) is forced to move on the blue circle. What does the transformed point do? It moves on an ellipse, whose size and orientation are determined by the actual transformation. At this point matrix methods become very handy.(though the 2D matrix methods cannot deal with translations)

gif for lin affine trans2

All this was constructed with my geometrical construction program (APP if you like) called GEOSTRUCT and available as a free web based application from


The program produces a listing of all the actions requested, and these are listed below for this application:

Line bb moved to pass through Point A
New line cc created, through points B and C
New Point D
New line dd created, through Point D, at right angles to Line aa
New line ee created, through Point D, at right angles to Line bb
New line ff created, through Point D, parallel to Line cc
New point E created as the intersection of Line ff and Line aa
New line gg created, through Point E, at right angles to Line aa
New line hh created, through Point B, at right angles to Line bb
New point F created as the intersection of Line hh and Line gg
New line ii created, through Point F, parallel to Line cc
New point G created as the intersection of Line ii and Line aa

G is the X coordinate, from X = x + y + 1 (added by me)

New line jj created, through Point G, at right angles to Line aa
New line kk created, through Point D, at right angles to Line cc
New point H created as the intersection of Line kk and Line bb
New point I created, as midpoint of points H and B
New line ll created, through Point I, at right angles to Line bb
New point J created as the intersection of Line ll and Line jj

J is the Y coordinate, from Y = -x/2 + y  (added by me)
and K is the transformed point (X,Y) Point J chosen as the tracking point (added by me)

New Line mm
Point D moved and placed on Line mm



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Filed under algebra, conics, construction, geometrical, geometry app, geostruct, math, ordered pairs, rigid motion, teaching, transformations, Uncategorized

“Shear”, the forgotten transformation.

Transformations of the plane are many and various.
The “nice” ones are “rigid motions”, and this term includes rotations, reflections and translations. The shape and the size of a geometrical object are not altered by a rigid motion.
There are also “shape preserving” transformations, called “dilations”, in which an object is stretched or shrunk equally in all directions.
An often overlooked transformation is the “shear”, in which there is a fixed line, and points not on that line are pushed parallel to the line in proportion to their distance from the line. Think of a stack of paper,perfectly stacked, and then pushed sideways so that the side of the pile is still flat. You will see a parallelogram at the front of the pile.
A shear will change the direction of a line, turn a rectangle into a parallelogram and turn a circle into an ellipse,
the area of any closed figure does not change at all.

Here is the static picture of a fixed point J, a fixed line, the x-axis, and a set of points on the horizontal line through A.
Also two triangles, LND and LDF, which are going to be sheared
shear transformation in xy plane
And here is the shearing in action, for varying amounts of shear, determined by the value of k.
gif for shear
Notice that triangle LMN changes a lot, and its area changes, but the areas of triangles LND and LDF do not change at all.
Not shown is a rectangle and a circle, which would change into a parallelogram and an ellipse, but their areas will not change with a shear.

For more on this go back to my Christmas post:

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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.

help pic 1
Two lines, with points placed on them
help pic 3
Three random lines with two points of intersection generated
help pic 6
Five free points, three generated circles and a center point
help pic 7
Three free points, connected as point pairs, medians generated
help pic 5
Two free circles and three free points, point pairs and centers generated
gif line and circle
GIF showing points of intersection of a line with a circle
hypocycloid locus
Construction for locus of hypocycloid
circle in a segment
GIF showing a dilation (stretch) in the horizontal direction
gif piston cylinder
Piston and flywheel
gif touching2circles
Construction for circle touching two circles
gif parabola
Construction for the locus of a parabola, focus-directrix definition.






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Quadrilaterals – a Christmas journey – part 3

So what else do stretches and shears do?
A stretch will turn a square into a rectangle if it acts in the direction of one of the sides.
A stretch will turn a square into a rhombus if it acts in the direction of a diagonal.
A shear will turn a square or a rectangle into a parallelogram.
Try it out !

Now to continue the journey – the mathematician now thinks “Is that it ? Are these the only transformations of the plane that map straight lines to straight lines ?”. His answer, with a Eureka moment, is “No ! What about those artists, with their perspective drawings ? Not only do straight lines in reality go to straight lines in their pictures, but when the lines are parallel in reality they go to convergent straight lines in the picture. “This is projection !”, cries the mathematician, and pursues the matter further and further…..

old master picArchitect_drawing_conceptual_2012

If I fix a sheet of glass or acrylic, sit still with marker pen in hand, and copy onto the glass exactly what I see through the glass I get a point projection of reality on the glass. Doing this when reality is a flat wall creates a projection from the reality plane to my glass plane. This is a new type of line preserving transformation of the points on a plane.

What is really nice about projection is that all the transformations we have seen so far can be described by projections.

First we have to classify projections, from a source plane to a target plane:
1. Point projection. The projection lines all pass through a fixed point (that’s the point where your eye was earlier). then each projection line passes through a point on the source plane, and where it hits the target plane is the transformed or projected point.
2. Line projection. The projection lines all start on a fixed line and are at right angles to that line. Imagine a spout brush or a very hairy caterpillar.
3. Parallel projection, in which the projection lines are all parallel to each other, as in rainfall or a bundle of spaghetti.

Here are translations, reflections, stretches and shears as projections:

Projection 0 Projection 1 Projection 2 Projection 3

What about rotations, you ask. Simply, a rotation can be seen as a sequence of two reflections.

But in general projection we lose the preservation of ratio. For example, the midpoint of a line segment is not projected to the mid point of the projected line segment:
A slice will do …..

midpoint 1midpoint 2It gets worse ! In the lower picture, as the point C is moved to the right along the line its image H moves further and further along, and, when C reaches D, H disappears altogether.
So if we look at the source plane from above, and we have a pair of lines that meet at point D, their projections must be parallel lines. (There could now follow a digression on the meaning of the word “infinity”, but it ain’t happenning).

What else ? Well, we have seen that ratios of distances are altered by projection, but when we take four points A, B, C and D on a line we can take two distance ratios (involving all four points) and take the ratio of these ratios. THIS quantity is NOT destroyed by projection. It is called the “cross-ratio” of the four points, and its value is (AB/AD)/(CB/CD), easier as (AB.CD)/(AD.CB)
(proof later)
A quick check:xratio testCopy the diagram to scale, but you can put the target line where you like.
Then measure EF, FG and GH, and calculate the cross-ratio. Should be the same, if you kept to the point matching.

Finally, and back to the quadrilateral:
Each diagonal has four points on it, two vertices and two intersection points with the other two diagonals.
If we join D to H, the point of intersection of the two diagonals BC and EF, we can see that the two sets of four points are connected by a projection from D, as H to H, B to F, G to I and C to E.
Consequently they have the same cross-ratio.

Not only that, it can be shown (one day !) that the value of this cross-ratio is -1, for all quadrilaterals, all the time.
Also, it is possible to map any quadrilateral to any other by a sequence of projections.

This is my introduction to projective geometry, a very interesting and underexposed branch of geometry. There may be a part 4 eventually !

And two more 3D gifs :

gif cube 3gif torus 2

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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)
translation v moving axes

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 ! ! !

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Lament to the Common Core geometry

Could I move this trapezoid
To that one, in the endless void?
I tried translation and rotation.
Then I had a crazy notion.
I would pass a rigid motion.
Result – a lovely hemorrhoid.

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Geometry test. Maybe simple ! Try it.

Three maybe parallel lines, AD BE CF
Two transversals, AC DF
question how many parallel
Given that AB is equal to BC and DE is not equal to EF
(“equal” = “congruent” if you like)
show that at most two of the maybe parallel lines can be parallel.
(proof by contradiction is a last resort)

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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
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

congruent line segs rigid transf

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:


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

 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!


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