Invariance

Question 1

What point is ALWAYS AN INVARIANT POINT IN ANY TRANSFORMATION
Huge importsnt trick question don’t lack

Answer 1

0,0, under every transformation, it stays there

Question 2

What is an invariant kine

Answer 2

This is where point on the line stay ON THE LINE after transformation, but jot necsdailt same position

Question 3

What is an invariant line of POINTS

Answer 3

Each point in the line will remain on the same point on the line

Question 4

What is a reflection line always in terms of invariance

Answer 4

A line of INVARIANT POINTS, points ok then NEVER MOVE POSITON

Question 5

How to determine whether a matrix trandfosmtion has any invariant point of lines

Answer 5

Applying the trandfosmtion to x y should give you same x y

Now see if the top and bottom are consistent, if they are then the kine of invariant ooints is that, if not they don’t have any

Question 6

How to find invariant lines?

Answer 6

Invariant line assume form y =mx +c

So any general point will be x, mx +c
So multiply trandfosmtion by this

Now in fact this point will be on the line y = mx +c

So equate
Rearrange equate to 0

, for it to be equal to 0 both parties in brackets must be 0, or c 0

Find what m can be, and if that value allows c to be anything or not

Might get two invariant lined, or one general

Question 7

If you found the invariant point of lines and now finding invariant lined, nice way ti check?

Answer 7

Well ina riantpoint if linedis a invariant line

So if one of your invadiantlined is thr invariant point of lines then it matches up!

Question 8

When you show why the trandfosmtion has no invariant points what to do

Answer 8

Find jf it has any invariant point of lines, as all invariant points lie there

Oh look the equations arentconsistent

Therefore the only point one lies is 0,0 for them to be consistent , so this the only point that is INVARIANT

Remember, the other method is to find invarisnt lines, where the points arent inabrainftbf

Question 9

On a reflection, where is the invariant point of lines. And where are the INVARIANT LINES

IMPORTANT

Answer 9

Invariant point of lines = reflection line

Invariant lines = lines that are PERPENDICUALR to the reflection line, as all these points will be reflected onto themselves!

So the line would be y = -grsdient of reflection line x + c

So many

Question 10

For a SHEAR, where is the invariant line of points and where are the invariant lines?

IMPORTSNT!

Answer 10

Invariant line of pointd = hsear line = the axis we keep CONSTSNT

All lined PERPENDICULAR TO THIS LINE ARE INVARIANT LINES, because all valued move parallel to this axis, so they remain on the same line,

So line equation would be y = grsdient of shear lime (as parallel) x + c ofc