Invadaince Flashcards
3 scenarios
Invariant point = point is mapped on to same point after transformation like anything to 00
2) invariant line of points
- all points on a line stay at exactly same points on line
- so like reflections , and then any point in a line will reflect on to same line
3) invariant line
- all points on a line remain on a line, but like can be moved across
- for example , translations and STRETCHES ETC!
Determine if point is invariant
Map it to a point if same then yh
2) determine if line of invariant point s
Map to xy gives xy
If two equations are consistent and give same line, then this line has all invariant points
How to find invariant lines?
Assume solutions lie on line y=mx+c
So mapping transformation to x y will give x’ y’ which is still on the line y= mx +c
So find x’ y’ and sub into y=mx+c in terms of m
Rearrange to = 0, in this case both terms need ti be 0
First term 0, x can’t me so m has to, and find the solutions
Second either c is 0 or bracket, find solutions
Now see when you sub in ine, if both brackets are 0, then c can be anything
So you have y=mx+c
But if only one bracket 0, then c has to be 0 to make it all 0
Thus its y=mx
Show thwt no other invariant points
Lit sub one into other and show that they only equal when x and y are 0!!!
This means only point is 00
For shears and invariant lines, where will all invariant lines be
Where is the INVARIANT LINE OF POINTS
As shears translate all points parallel to the shear line of invariant points, all points that are parallel to this line stays on their line
So all invarjant lines are y= (gradient of shear line ) £c
2) invariant line of points IS THE SHEAR LINE , the axis we keep constant
Where are the invariant lines in a reflection?
Where is the invariant line of points?
Perpendicular to the line of reflection so grsdient is negative reciprocal and then just +c
So there’s infinite amount
Reflection line = invariant line of points
Now if y=2x is a solution, can sub in to what
Y is 2x
