Find Point At Edge of Rect, Given Angle

[Swivelgames]

Known variables: h, w, a, b, c¹, d¹, e¹, f¹, and z°
Unknown variables: c², d², e², f², m, and n

I need to find m,n and then translate c¹,d¹ to c²,d² and the same for e¹,f¹ to e²,f².

Other fun facts:
a = w/2
b = h/2

This is a real-world problem, so it was one that I had to come up with the steps. I'm creeping through this problem, and now I'm stuck. I got z using trigonometry dynamically, however, I'm having trouble with the rest of this problem. Can anyone help me through the rest of the steps? I know what I need, but I don't know the equations or steps to get the unknown variables.

(Honestly, I'm also looking to calculate the distance between (c²,d²) and (m,n) as well as the distance between (e²,f²) and (m,n) as the actual result, but I know the equation for those last steps).


Yay example values:

• Rectangle w × h
  1440 × 434
  
• Center Point (a,b)
  (720,217)
  
• Point (c¹,d¹)
  (918,177)
  
• Point (e¹,f¹)
  (1018,277)
  
• Angle z°
  135°

I forgot to mention, you can see the axes in the image above, but you should note that the Y axis is inverted. The higher up Y, the lower the coordinate.

[Swivelgames]

I've attempted to make the problem a little easier to read and work with. The ideal solution will work for the first example, as well as the following example.



Known: A, B, C, P, R, T, Z, h, w
Need: D, E, F; Length of dcef and ef segments

NOTE: I was unable to post this using the "angle" symbol, so the less-than sign will have to suffice. Let me know if it gets confusing at all!

Known Data:
h = 600
w = 800

C is at (w/2,h2) = (400,300)
R is at (400,0)
P is at (400,600)

A is at (534,88)
B is at (740,296)
T is at (534,0)
Z is at (534,600)

<TAB = 135°
<ZAB = 45°

<RCE = <TAB
<PCE = <ZAB

Problem Solving
<ZFE = 90 - <PCE
<ZFE = 45°

segCP = segRC = 400
segCF = sin(<PCF) × segCP
segCF = sin(45) × 400
segCF = 282.8427125


Now all I really need is the length of segEF to solve the problem, since we know that segDE = segAB.

So segDF = segDE + segEF

Can anyone help me get segEF?

Something to note: This is a real-world problem where <TAB will be anywhere from 0° to 360°, which in-turn would affect what side and position F would be. And <RCF always = <TAB

[Swivelgames]

Hello...? >.>

[maggiemagnet]

Is the line through (a, b) and (m, n) parallel to the line through (c₁, d₁) and (e₁, f₁)? When you say "translate", are you meaning that you need to find the points on the first line (the one through the center point) that are also on the perpendiculars through the points on the second line? Thanks!

[Swivelgames]

Yes, those lines are parallel.

To understand what I mean by translate a little better, let me explain the situation

I start out with points (c1,d1) and (e1,f1). They make a (in the example) 135* angle based on a line from (c1,d1) to the top of the rectangle. Next, I want to "translate" (c1,d1) and (e1,f1) to be on a parallel line in the center of the rectangle with the same angle. Now, I have to figure out how long the segment of (e2,f2) and (m,n) is to solve the problem. So, by translate, I simply mean move segment c1d1e1f1 to the line abmn.



In the second example I gave (a bit easier to work with), I start out with points T, A, B, and C. I want to translate line AB to a parallel line intersecting point C. Then, I want to translate line AB to the line that intersects at point C. That new segment is DE. Points A, B, D, and E need to create a proper rectangle, where all sides (including AD and BE) are parallel. Now, I just need to find the length of EF.


Also, I wanted to note a couple things from other people's questions.

<TAB will be anywhere from 0° to 360°, which in-turn would affect what side and position F would be.

<RCF always = <TAB

E is not a point on AZ

A may change coordinates, DE will always be parallel and congruent to AB

[maggiemagnet]

What if A and B are placed so that B, say, would be outside of the rectangle after translation to the parallel line through the center?

[Swivelgames]

Indeed, point A, B, D, and E can be outside of the rectangle. In which case, T will also be outside the rectangle. T is always directly above A.

[Swivelgames]

bump. Still lost! :'(

[nona.m.nona]

It is not clear why the measure of angle Z would be necessary, or indeed trigonometry. To locate the translated points, regard the center point as being at the origin, for simplification. Find the slope through the two original points. From this, find the perpendicular slope.

Find the equations of the perpendicular lines through the two original points; find also the equation of the parallel line through the "origin". Using systems of equations, find the intersection points of the perpendiculars with the parallel. These intersection points are the translated points desired.