Started by Aug 06 2012 04:24 AM

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5 replies to this topic

Posted 06 August 2012 - 04:24 AM

Hello everyone,

I have the points for a vehicle trajectory in 3D space now I want to make a triangle strip that shows the vehicle trajectory.

Like in this image I have the line, now I need the strip.

My main question is how to calculate the x(width) and z(depth) since depending on slope it's not as simple as adding half the width of the segment.

Since I have two points in the line and the slope, I should be able to calculate what is the point at D distance from the line points. I just can't put my finger on the math.

Can anyone help me out?

I have the points for a vehicle trajectory in 3D space now I want to make a triangle strip that shows the vehicle trajectory.

Like in this image I have the line, now I need the strip.

My main question is how to calculate the x(width) and z(depth) since depending on slope it's not as simple as adding half the width of the segment.

Since I have two points in the line and the slope, I should be able to calculate what is the point at D distance from the line points. I just can't put my finger on the math.

Can anyone help me out?

Posted 06 August 2012 - 10:43 AM

I think the reason that you are struggling is mainly because you do not have sufficient information on the trajectory. Besides the direction of the line, there is a freedom of movement in the rotation around the line. For instance, you could imagine a plane doing a "barrel roll" while staying on the same trajectory line, resulting in a nice double helix like effect for what you want to achieve.

Seeing as how you want to follow a vehicle, I would advice storing the complete transformation per point, instead of just the position. If you only want to reproduce this effect, then adding a "side" vector would be sufficient.

Imagine adding a small XYZ axis system on each point**p**_{i} in your trajectory, where one vector **f** points in the direction the vehicle is headed, one vector **u** points up from your vehicle and the third vector **s** points to the left (or right) side of the vehicle. Ideally, these vectors should be normalized. If you have trouble obtaining them, please let us know.

Given these vectors, finding the points along the sides of your trajectory is suddenly trivial:**L**_{i} = **p**_{i} + a **s**, and **R**_{i}** = ****p**_{i} + a **s**, for Left and Right, where a is half the width of the trajectory.

The order for the triangle strip would then look something like:**L** _{0} - **R**_{0} - **L**_{1} - **R**_{1} - **L**_{2} - **R**_{2} - etc...

Hope it's a little clear.

Seeing as how you want to follow a vehicle, I would advice storing the complete transformation per point, instead of just the position. If you only want to reproduce this effect, then adding a "side" vector would be sufficient.

Imagine adding a small XYZ axis system on each point

Given these vectors, finding the points along the sides of your trajectory is suddenly trivial:

The order for the triangle strip would then look something like:

Hope it's a little clear.

**Edited by Ignifex, 06 August 2012 - 10:44 AM.**

Posted 07 August 2012 - 09:37 AM

Thank you so much for your answer!

I have a few questions though. Sorry if they are very basic.

The data I have acess to is (using your notation) Pi, Pi+1 and Phi or the Roll angle.

I know that f = (Pi+1)-Pi

For u, the idea was to rotate the unitary y vector according to Phi. But I don't remember how to apply a rotation to a vector.

Finally, for s, I was planning to calculate the normal of f and u.

How do I rotate?

I have a few questions though. Sorry if they are very basic.

The data I have acess to is (using your notation) Pi, Pi+1 and Phi or the Roll angle.

I know that f = (Pi+1)-Pi

For u, the idea was to rotate the unitary y vector according to Phi. But I don't remember how to apply a rotation to a vector.

Finally, for s, I was planning to calculate the normal of f and u.

How do I rotate?

Posted 07 August 2012 - 01:23 PM

When approaching the rotation through angles, the yaw-pitch-roll system is probably the most intuitive. Once you know these angles, finding any vector is relatively easy.

As you mentioned, you already have a roll angle. The yaw and pitch can then be computed from your forward vector**f**, using it's x, y and z components. This is a little rough and you should be careful when the x and z components are both 0, but the computation is as follows:

yaw = atan2( -x, -z )

pitch = atan2( y, sqrt( x^{2} + z^{2} ) )

Given these angles, I believe it's fastest to compute**s** using some rotation matrices. For yaw-pitch-roll, the rotation is applied as YPR * vector, or ZYX * v. Given a simple vector (-1, 0, 0), it's components will be:

x = - (cos(yaw) *cos(roll) + sin(yaw) * sin(pitch) * sin(roll))

y = - (cos(pitch) * sin(roll))

z = - (cos(yaw) * sin(pitch) * sin(roll) - sin(yaw) * cos(roll))

Note that I assumed you use the OpenGL eye space coordinate system, since you mentioned y is pointing up. I follow the right hand rule for the rotations, also the roll. A (small) positive roll angle turns a plane left, a positive pitch goes up and a positive yaw is also left.

As you mentioned, you already have a roll angle. The yaw and pitch can then be computed from your forward vector

yaw = atan2( -x, -z )

pitch = atan2( y, sqrt( x

Given these angles, I believe it's fastest to compute

x = - (cos(yaw) *cos(roll) + sin(yaw) * sin(pitch) * sin(roll))

y = - (cos(pitch) * sin(roll))

z = - (cos(yaw) * sin(pitch) * sin(roll) - sin(yaw) * cos(roll))

Note that I assumed you use the OpenGL eye space coordinate system, since you mentioned y is pointing up. I follow the right hand rule for the rotations, also the roll. A (small) positive roll angle turns a plane left, a positive pitch goes up and a positive yaw is also left.

**Edited by Ignifex, 07 August 2012 - 01:57 PM.**

Posted 08 August 2012 - 02:10 AM

I see.

I actually have yaw-pitch-roll, just didn't know it was easier.

I'll use Pi for location and your system for the vectors.

Is this written in any kind of tutorial/book/crash course?

I actually have yaw-pitch-roll, just didn't know it was easier.

I'll use Pi for location and your system for the vectors.

Is this written in any kind of tutorial/book/crash course?

Posted 08 August 2012 - 04:01 AM

What I have described is all Linear Algebra and the use of rotation matrices. You can pick up any introduction level book on the topic, which should also include transformations as a part of matrix multiplication.

As for online sources, wikipedia might be a good place to start. Wolfram is good reference material, but it might not have the best introductions.

For real world applications such as this one, it usually helps to make a drawing of what you are trying to do. Draw the vehicle as it heads in a all 0 angle direction, then consider what rotations should be applied and in what order. Rotations in 3D is often a bit of a pain, but it gets better with some practice.

Also, using a math library will really help to clean up your code and you can avoid writing out rotations by hand like I showed.

As for online sources, wikipedia might be a good place to start. Wolfram is good reference material, but it might not have the best introductions.

For real world applications such as this one, it usually helps to make a drawing of what you are trying to do. Draw the vehicle as it heads in a all 0 angle direction, then consider what rotations should be applied and in what order. Rotations in 3D is often a bit of a pain, but it gets better with some practice.

Also, using a math library will really help to clean up your code and you can avoid writing out rotations by hand like I showed.