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Topic: Trigonometry help: line corners

maximile
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posted July 05, 2006 09:40
I've been messing around with OpenGL programming, and I came across a problem with its drawing of lines.
(I'm using a lot of pics here, so I guess lots of you won't see this. But I can't describe it properly with text.)
Say I fed it the following three points:
If I ask it to render it as a line with a thickness, it'll do it as two rectangles, meaning that you get an ugly overlap. Reading around, it seems that this is a problem with OpenGL, and I can't find any help on the topic.
So I'm going to have to use triangles, and split the line like this:
The problem I'm having is finding the two points I've marked with an arrow. The other points are easy enough  although I've had to use trig functions, and I get the feeling that there's a better way perhaps using just the gradient. Here's how I did it:
To find the point marked 'a', I used thefollowing equation:
The others can be found just as easily by swapping some signs and so on. So I'm only struggling with the two in the middle.
Can any of you trig gurus help me? I've tried and failed, and I couldn't even find anything on Google (although I didn't really know what to search for).
Here's another diagram, at a more acute angle. (Oops... I reused the letter a. Well, it's a different one this time.)
So points a and b are the ones to find. I obviously need to find where the outside lines meet (for point a) and where the inside lines meet (for point b). But I can't work out how to express this in terms of the three points from the first diagram.
Any help would be appreciated, as ever. Feel free to tell me I'm stupid, or that I've missed something obvious, or that the question doesn't make sense.
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TheMoMan
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posted July 05, 2006 10:43
maximile________________________I used to think I was a geometry wizz untill your problem, now I am going huh. Are you looking for the third line from three ordered points? a(x,y) b(x,y) c(x,y) if so there should be help on the web, as to programing to find the line I would be lost.
 Those who would give up essential liberty to purchase a little temporary safety deserve neither liberty nor safety.
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Metasquares
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posted July 05, 2006 11:38
I understand what you're trying to do, but I'm having a hard time interpreting the diagrams.
Theta's the angle between something and the Xaxis, but what? It looks like it's a point midway between (x,y) and (x1,y1).
Also, the images you're drawing look 3D  it makes it confusing, since it looks like there's a Z axis. Since you're just dealing with a 2D triangle, it may help to draw the diagrams in 2D.
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canadiangeek
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posted July 05, 2006 18:28
Max, I think I understand what you're trying to do... but am currently doing a lab writeup. While, under normal circumstances, I'd happily use this as an excuse to procrastinate, I just don't have time (I think this'd be a page or so of algebra).
If someone else hasn't solved it when I get back from my lectures tomorrow, I'll try and solve it for you.
 whenever you build something that's idiotproof, someone comes out with a better idiot
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Ivan
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posted July 05, 2006 23:30
This shoudl work, but it's early in the morning and i could be dead wrong.
find equation of line connecting (x1,y1) and a. find equation of perpindicular line at a. (Call this equation x)
find equation of line connecting (x2,y2) and (x2+t, y2+t) (assuming they're vertically collinear, which it looks like they are). then find the equation of the line perpindicular to that at point (x2+t, y2+t). (call this equation y)
set x=y to find their intersecting coordinates.
do the same to find the inside corner, but use the other points in finding your equations.
lengthy and roundabout, but it should work.
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maximile
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posted July 06, 2006 03:39
Thanks for the interest, everyone.
Are you looking for the third line from three ordered points? a(x,y) b(x,y) c(x,y) if so there should be help on the web, as to programing to find the line I would be lost.
No, I'm just looking for an equation that will get the marked points for given values of x, y, x1 etc. Programming shouldn't come into it; I'm happy to convert an equation into code myself.
Theta's the angle between something and the Xaxis, but what? It looks like it's a point midway between (x,y) and (x1,y1).
Theta is the angle between the xaxis and the line (x1,y1) to (x,y). It doesn't matter whether it's midway or not. I work out theta just using an inverse tangens (modified to work properly in four quadrants).
Also, the images you're drawing look 3D  it makes it confusing, since it looks like there's a Z axis. Since you're just dealing with a 2D triangle, it may help to draw the diagrams in 2D.
True... here's a (hopefully) clearer version.
find equation of line connecting (x2,y2) and (x2+t, y2+t) (assuming they're vertically collinear, which it looks like they are). then find the equation of the line perpindicular to that at point (x2+t, y2+t). (call this equation y)
Thanks for the help. However, the solution needs to work with any three points, not just the current ones. Also, I don't know how to find the equation of a line.
What I really need is something like I have for finding point a  an equation in which the x and y values are isolated, and expressed in terms of the x and y values of the other three points. Or theta, or any angle between any of the points, because those are easy to work out.
Feel free to ask for more information if any of you are still interested but confused, and thanks again for the help so far.
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Metasquares
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posted July 06, 2006 05:52
I think I may have figured it out.
Let's assume for a moment that theta is 0. If this is the case, you'll just get a rectangle with a width of t.
Point (x,y) will be located at the midpoint between unknown points a and b. Since the length of this side is t, that means the vertical distance between a or b and y is t / 2. In this case, they also lie on the same xcoordinate.
So a will have the coordinates (x, y  t / 2) and b will have the coordinates (x, y + t / 2) before rotation.
Now we stop assuming the value of theta and apply a rotation transformation. The formula for this is x' = x cos(theta) + y sin(theta), y' = x sin(theta) + y cos(theta).
So for point a, we should get (x cos(theta) + (y  t/2) sin(theta), x sin(theta) + (y  t / 2) cos(theta)).
Point b is the same save for flipped signs before the t / 2 terms.
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Ivan
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posted July 06, 2006 09:31
quote: Thanks for the help. However, the solution needs to work with any three points, not just the current ones. Also, I don't know how to find the equation of a line.
y= (y2y1)/(x2x1)
but uh, other than that i choose not to help you, because that requires a little more thought than i feel like investing in this. sorry.
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Metasquares
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posted July 06, 2006 09:54
quote: Originally posted by Ivan: y= (y2y1)/(x2x1)
You mean m=(y2y1)/(x2x1). You still need to find the yintercept to get the equation of the line.
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maximile
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posted July 06, 2006 12:06
Thanks everyone  I'll try this tonight, and if I'm successful I'll let you know.
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Ivan
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posted July 06, 2006 12:20
quote: Originally posted by Metasquares: quote: Originally posted by Ivan: y= (y2y1)/(x2x1)
You mean m=(y2y1)/(x2x1). You still need to find the yintercept to get the equation of the line.
You're absolutely right. that's what I get for posting withing a minute of waking up.
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maybe.logic
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posted July 08, 2006 02:21
quote: Originally posted by Ivan: quote: Originally posted by Metasquares: quote: Originally posted by Ivan: y= (y2y1)/(x2x1)
You mean m=(y2y1)/(x2x1). You still need to find the yintercept to get the equation of the line.
You're absolutely right. that's what I get for posting withing a minute of waking up.
[rant] everytime sombody makes an error with maths/physics it is always an exuse along simillar lines...
Oh I jst woke up, Oh it was really late, Oh i was so pissed... and it goes on.
[/rant]
I'm sure I have done it from time to time as well, i just think its funny.
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