Triangle Area Second Derivative [Check my math]
Author
With a triange {r1, r2, r3}
I need to calculate the 2nd derivative of the triangle's area.
Let hi and wi be the height and width respective with ri...
Is it true that d2/dt2 (Area(t)) =
1/2 * (
h21*w1+h22*w2+h23*w3).
where h21 for example is the second derivative of h1 with respect to time..
Define E1 = r2 - r1 and E2 = r3 - r1. The area is A = Length(Cross(E1,E2))/2. Multiply by 2 and square to get 4*A^2 = (E1xE2).(E1xE2), where "x" means cross product and "." means dot product.
Compute the first derivative A', 4*A*A' = (E1xE2).(E1xE2' + E1'xE2), which you can solve for A'.
Compute the second derivative A", 4*(A*A" + A'*A') = (E1xE2).(E1xE2" + 2*E1'xE2' + E1"xE2) + (E1xE2' + E1'xE2).(E1xE2' + E1'xE2), which you can solve for A".
Compute the first derivative A', 4*A*A' = (E1xE2).(E1xE2' + E1'xE2), which you can solve for A'.
Compute the second derivative A", 4*(A*A" + A'*A') = (E1xE2).(E1xE2" + 2*E1'xE2' + E1"xE2) + (E1xE2' + E1'xE2).(E1xE2' + E1'xE2), which you can solve for A".
Be careful to NOT give a complete answer to problems such as this, which are academic in nature. The poster asked to check his work, NOT do the problem for them, so the post is OK! But, to produce a full answer to an academic-style post can result in a thread being closed by forum policy. Forum FAQ
The followup message isn't completely helpful, since it does not suggest how to compute the derivatives of E1' and E2' with respect to time (the independent variable given by arithma).
Arithma, could you clarify hi and wi a bit. For example, is h1 the perpendicular distance from vertex r1 to the opposite edge, and w1 the width of the opposite edge (w1 = length of the vector from r2 to r3)? I'm not entirely clear how you are defining height/width. Also, I think there must be something wrong since you do not have any time derivatives of the width.....
Hopefully there are some clues here. Let us know how you progress.
The followup message isn't completely helpful, since it does not suggest how to compute the derivatives of E1' and E2' with respect to time (the independent variable given by arithma).
Arithma, could you clarify hi and wi a bit. For example, is h1 the perpendicular distance from vertex r1 to the opposite edge, and w1 the width of the opposite edge (w1 = length of the vector from r2 to r3)? I'm not entirely clear how you are defining height/width. Also, I think there must be something wrong since you do not have any time derivatives of the width.....
Hopefully there are some clues here. Let us know how you progress.
Quote:Original post by grhodes_at_work
Be careful to NOT give a complete answer to problems such as this, which are academic in nature. The poster asked to check his work, NOT do the problem for them, so the post is OK! But, to produce a full answer to an academic-style post can result in a thread being closed by forum policy. Forum FAQ
Close the thread if you must. I am not skilled in judging whether a post is "legitimate". If this person is posting a homework question and plans to use the followups as the answer to submit to a professor, then the professor should have the responsibility of questioning the student to see if he/she really does know the answer. I just do not understand gamedev.net's insistence on being the Academic Police.
Quote:
The followup message isn't completely helpful, since it does not suggest how to compute the derivatives of E1' and E2' with respect to time (the independent variable given by arithma).
The poster never said how the vertices vary with time. If he/she knows how to compute a derivative, then the followup message is completely helpful.
Author
Thanx Wasting_Time for ur reply, but I really need to check whether I did is valid or not as it results in a form that am comfortable with (ready to implement).
As for grhodes_at_work, your interpretation of hi and wi are correct.
To come up with this formula I seperated some variables first:
Let 2nd deriv of r with respect to t be a... {t|i=1 to 3} is a set of unnecessarily-dependent variables (not dependent).
Partial 2nd derivative of Area with respect to t is 1/2*h(2nd deriv wrt t)*w
The w(2nd deriv wrt t) is zero since it does not change with respect to t...
Now set all t = t, 2nd derivative of area with respect to t is simply a chain rule which envolves a sum over i of some simple sort that will lead to the formula that I concluded earlier.. Note that am still unsure of what I did, especially with the partial derivative thingy and width's 2nd derivative.
As for being a homework of some sort, I assure you it is not. This is part of a 2d spring model that works with triangles. You can check this previous post of mine to be 100% sure about it.
As for grhodes_at_work, your interpretation of hi and wi are correct.
To come up with this formula I seperated some variables first:
Let 2nd deriv of r with respect to t be a... {t|i=1 to 3} is a set of unnecessarily-dependent variables (not dependent).
Partial 2nd derivative of Area with respect to t is 1/2*h(2nd deriv wrt t)*w
The w(2nd deriv wrt t) is zero since it does not change with respect to t...
Now set all t = t, 2nd derivative of area with respect to t is simply a chain rule which envolves a sum over i of some simple sort that will lead to the formula that I concluded earlier.. Note that am still unsure of what I did, especially with the partial derivative thingy and width's 2nd derivative.
As for being a homework of some sort, I assure you it is not. This is part of a 2d spring model that works with triangles. You can check this previous post of mine to be 100% sure about it.
Quote:Original post by arithma
The w(2nd deriv wrt t) is zero since it does not change with respect to t...
Ah, yes, I see that now!
Quote:Original post by arithma
As for being a homework of some sort, I assure you it is not. This is part of a 2d spring model that works with triangles. You can check this previous post of mine to be 100% sure about it.
OK, that's fine. I am a bit sensitive to this. There are folks who ask academic-style questions legitimately. I just disapprove of people, especially kids, who just want to not do the work for themselves---the policy is to strongly encourage people to struggle through these problems for themselves. Not that I felt this was coming from a kid, but I do go into automatic mode, :).
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