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…..
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:
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 …..
It 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)
A quick check:Copy 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 :