Comments for generalrelativity

Comment on Constrained Dynamics 1: Particle Constrained to Curve by drew

drew — Thu, 11 Mar 2010 17:08:10 +0000

@Ben
The issue with a Euler integrator is that it’s really inaccurate. You’re solving for the acceleration at a specific moment in time, then stepping forward over a time interval as if acceleration is constant. Because the Lagrangian formulation essentially creates a space where the constraint is always satisfied, there is no concept of an invalid position, velocity or acceleration, insomuch as the particle will always end up on the curve. So any oscillation you’re seeing is likely just a graphical quirk (though I don’t really notice anything!). That said, the integration problem appears not as a deviation from the curve, but as a change in energy. I note above that I had seen that energy appears to leak over time, but my intuition tells me it would grow–I’ll have to look at that again. Regardless, it shouldn’t be visibly noticeable within the context of your simulation as is.

Comment on Constrained Dynamics 2: Joints and Global Constraints by Santiago Puente

Santiago Puente — Wed, 10 Mar 2010 03:49:57 +0000

Pleeeeeaseeee do write an article about the velocity-based constraint solverrrrr!!!
Pleeeeeaseeee.
These articles are awesome.

Don’t stop!

SP.

Comment on Constrained Dynamics 1: Particle Constrained to Curve by Ben

Ben — Mon, 08 Mar 2010 23:27:17 +0000

@drew

Hey Drew,
many thanks for your explanation! My memory has come back now and of course my earlier approach was nonsense. I guess it was just random coincidence that my result looked close to the true path.

The following is how my own little version of your example looks like after above corrections:
http://www.dobschin.de/games/game-physics/lhmslide/LhmSlide.html

I’m using an Euler-integrator. Would it improve anything, if I used a more accurate one? For example, it looks like my particle is kind of oscillating about the wire while moving downward. But I’m not sure if I observe that correctly or if I just stared at my monitor for too long!?

Comment on Constrained Dynamics 1: Particle Constrained to Curve by drew

drew — Sun, 07 Mar 2010 17:00:13 +0000

Hey Ben,

Thanks for spotting that, it’s fixed now.

Your issue is that we’re not solving for acceleration in a cartesian frame, we’re solving for acceleration in the generalized coordinate, q. Having found q” we can solve for q’ and q by integrating this 1D value. The position of the particle is [q,(-q^2)/100]. So there’s no concept of an x and y component in the Lagrangian equations of motion–everything is expressed in terms of the generalized coordinate. Because position in the cartesian system is a function of q, the particle can NEVER leave the parabola. So looking at your setup, you have a symplectic Euler integrator. Replacing all the x’s and y’s with q’s:

var qDDot:Number = (0, -( -q * qdot * qdot - 50 * g * q ) / ( 2500 + q * q ));
qDot += qDDot; //qDot is your velocity
q += qDot;

//now transform into cartesian coordinates to solve for your particle’s position
bead.x = q;
bead.y = -( q * q ) / 100;

I’m planning on releasing all the source code I use, including these examples and the dynamics harness that runs the simulations once I get it cleaned up enough.

Hope that helps!

Comment on Constrained Dynamics 1: Particle Constrained to Curve by Ben

Ben — Sun, 07 Mar 2010 12:52:30 +0000

When trying to implement the Lagrangian approach in AS3 my particle leaves the curve slightly, but annoyingly increasingly. Can you give me a hint?
So far, every frame, I’m setting my particle’s y-component of acceleration to your equation:
var q:Number = bead.x;
var qdot:Number = bead.v[ 1 ]
bead.a[1] = (0, -( -q * qdot * qdot - 50 * g * q ) / ( 2500 + q * q ));
bead.a[0] = 0;
I use the following to change the particles position:
bead.v[0] += this.a[0];
bead.v[1] += this.a[1];
bead.x += this.v[0];
bead.y += this.v[1];
I’m completely new to this, is this the problem of the integrator you mentioned? How do I change my particles position every frame correctly?

Two suggestions, maybe for a later tutorial:
- Explain how you draw your parabola.
- Explain how you get your particle to move to and fro.
Many thanks!

Comment on Constrained Dynamics 1: Particle Constrained to Curve by Ben

Ben — Sun, 07 Mar 2010 12:17:44 +0000

Have been looking for something like that all over the net. Very nice tutorial!

Looks like I spotted a typo, though. $\partial T / \partial \dot{q} = m \dot{q} (\frac{1 + q^2}{2500})$ should be
$\partial T / \partial \dot{q} = m \dot{q} (1 + \frac{q^2}{2500})$, I think. Or am I missing something?

Comment on Collision Detection: Circle/Line Segment, Circle/Capsule by mirhagk

mirhagk — Thu, 18 Feb 2010 17:33:35 +0000

Thank you for this article. I literally spent all of last night trying to figure out circle-line collision detection but without the interwebs of course. Now I log on today and the first result is this amazing tutorial. I’m still going to have to do a lot of work translating this code into the syntax I’m using but thanks a lot drew. To be honest I don’t quite get why the code for the closest point on the line works, but I comprehend enough to get by

Comment on Constrained Dynamics 2: Joints and Global Constraints by drew

drew — Mon, 15 Feb 2010 13:33:59 +0000

Maybe my sinister plan is to trick people into reading math…

Comment on Constrained Dynamics 2: Joints and Global Constraints by makc

makc — Mon, 15 Feb 2010 10:44:13 +0000

you need some kind of spoiler tag to wrap “Global Form of the Constraint Problem” § saying “truckloads of math, open at your own risk”, so that people see simpler imlementation §§ first, and stay within their zone of comfort it’s a good thing you have included demo swf at the top.

Comment on New Flash Animation Library by drew

drew — Thu, 11 Feb 2010 17:56:18 +0000

@mga: Do you want each character to fit around the spiral, or just a single TextField? A few people have asked for an “align to path” option, I might add it in to the next release. Wouldn’t be too hard to hack it in yourself though!