Simple coordinate transformation question

  • Thread starter iScience
  • Start date
  • #1
465
4

Main Question or Discussion Point

http://i.imgur.com/MDigPh5.png


if i have my original coordinate (white) and i am transforming this into the red coord. , could someone explain to me why y=y'cos[itex]\phi[/itex] is incorrect and why y'=ycos[itex]\phi[/itex] is correct?
 

Answers and Replies

  • #2
HallsofIvy
Science Advisor
Homework Helper
41,791
919
Neither of those is correct unless you are just giving part of the formula- in which case both are correct! y' depends on both y and z and y depends on both y' and z'.

Given any point in the plane, drop perpendiculars from the point to the y and y' axes. The angle at the point is [itex]\phi[/itex], the distance from the point to the foot of the perpendicular to the y' axis is z' and the distance from the point to the foot of the perpendicular to the y axis is z. Similarly, the distance from the origin to the foot of the perpendicular to the y-axis is y and the distance from the origin to the foot of the perpendicular to the y'- axis is y'.

The distance from the foot of the y-axis to the intersection of that perpendicular is [itex]y tan(\phi)[/itex]. The length of the rest of that perpendicular is the hypotenuse of that right triangle and so is [itex]\frac{z'}{cos(\phi)}[/itex]. Then [itex]z= y tan(\phi)+ \frac{z'}{cos(\phi)}[/itex]. Multiply both sides by [itex]cos(\phi)[/itex] to get [itex]z cos(\phi)= y sin(\phi)+ z'[/itex] so that [itex]z'= z cos(\phi)- y sin(\phi)[/itex]. Similarly, [itex]y'= z sin(\phi)+ y cos(\phi)[/itex].

You can solve those two equations for y and z or simply replace [itex]\phi[/itex] with [itex]-\phi[/itex] (the opposite of "rotating through angle [itex]\phi[/itex]" is "rotating through angle [itex]-\phi[/itex]"). Since [itex]sin(-\phi)= -sin(\phi)[/itex] and [itex]cos(-\phi)= cos(\phi)[/itex] we can just change the sign in front of the sines: [itex]y= -z' sin(\phi)+ y' cos(\phi)[/itex] and [itex]z= z' cos(\phi)+ y' sin(\phi)[/itex].
 
Last edited by a moderator:

Related Threads for: Simple coordinate transformation question

  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
1
Views
2K
Replies
10
Views
907
Replies
1
Views
516
  • Last Post
Replies
6
Views
903
Replies
3
Views
6K
Replies
2
Views
877
Top