stoutfellow (![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png)
stoutfellow) wrote2006-01-18 06:06 pm
stoutfellow) wrote2006-01-18 06:06 pm
Entry tags:
DG, Part 2: Twist and, Well, Not *Shout* Exactly...
The story so far:
Any reasonably nice curve has, at each point, three quantities associated with it: the tangent vector, the principal normal vector, and the curvature. These describe the direction the curve is headed, the direction in which the curve is bending, and the speed with which it is bending. There are two more bits that need discussion.
The next piece of the puzzle is the binormal vector. This is a vector with length 1 which is perpendicular to both the tangent and the principal normal; furthermore, the three together form a "right-handed system". Take your right hand and hold it in front of you, palm up. Let your thumb point to the side, your index finger forward, and your middle finger straight up. (Depending on your sensibilities, you may prefer to let your other two fingers also point upward...) If you then put your hand so your thumb points in the direction of the tangent vector and your index finger in the direction of the principal normal, your middle finger will point in the direction of the binormal. (A "left-handed system" works similarly, but with your left hand.)
Something interesting happens at this point. It turns out that the rate of change of the binormal is always a multiple of the principal normal. We define the torsion at the point to be the negative of the ratio between the rate of change of the binormal and the principal normal. (We make it the negative for reasons which are essentially aesthetic.)
At each point of the curve, we have three perpendicular vectors, the tangent, the principal normal, and the binormal, each with length 1. We can envision them as defining a coordinate system, with the tangent vector pointing in the x direction, the principal normal in the y direction, and the binormal in the z direction. This system of coordinates turns out to be a very nice system - remember that ellipse! - at least as far as the neighborhood of that one point is concerned. The system changes as we move along the curve, but it does so in a smooth fashion; you can visualize the axes rotating and twisting as you slide along the curve. We call something like this a moving frame.
One last thing: Messrs. Frenet and Serret proved a nice little theorem about their apparatus. We know, from the definitions of the curvature and torsion, what the rates of change of the tangent vector and the binormal are; what about the rate of change of the principal normal? It turns out that the rate of change of the principal normal is the torsion times the binormal, minus the curvature times the tangent. Represented mathematically, this is beautifully symmetric (well, technically, it's antisymmetric). It looks like this:
For the next post, I'll turn my attention from curves to surfaces.
Addendum: Erk. There are a couple of other details about curves that I should probably talk about, and maybe do an example. Surfaces after that.
Any reasonably nice curve has, at each point, three quantities associated with it: the tangent vector, the principal normal vector, and the curvature. These describe the direction the curve is headed, the direction in which the curve is bending, and the speed with which it is bending. There are two more bits that need discussion.
The next piece of the puzzle is the binormal vector. This is a vector with length 1 which is perpendicular to both the tangent and the principal normal; furthermore, the three together form a "right-handed system". Take your right hand and hold it in front of you, palm up. Let your thumb point to the side, your index finger forward, and your middle finger straight up. (Depending on your sensibilities, you may prefer to let your other two fingers also point upward...) If you then put your hand so your thumb points in the direction of the tangent vector and your index finger in the direction of the principal normal, your middle finger will point in the direction of the binormal. (A "left-handed system" works similarly, but with your left hand.)
The binormal vector B is defined to equal TxN. Since T and N have length 1 and are perpendicular, B also has length 1.Now, here's the thing about the binormal. Suppose your curve lies in a plane. Then (starting from the point you're looking at) the tangent vector and the principal normal both lie in that plane. That means that the binormal must be perpendicular to that plane, and there are only two length-one vectors perpendicular to any given plane. It turns out that in that case the binormal must be constant - it has to be one of those two vectors, and the same one no matter which point you're looking at. The converse is also true; if the binormal is constant, the curve lies in a plane. To phrase it differently: the rate of change of the binormal measures the tendency of the curve to twist out of a plane.
Something interesting happens at this point. It turns out that the rate of change of the binormal is always a multiple of the principal normal. We define the torsion at the point to be the negative of the ratio between the rate of change of the binormal and the principal normal. (We make it the negative for reasons which are essentially aesthetic.)
Apply the product rule to the definition of B: B'=T'xN+TxN'. But T' is a multiple of N, so that first term is zero. Thus B'=TxN', so it must be perpendicular to T. It's also perpendicular to B, because B has constant length 1 - see the argument in the previous post about the derivative of T. Since B' is perpendicular to T and B, it's a multiple of N. The symbol for the torsion is τ; it is defined by the equation B'=-τN.These five quantities - the tangent, principal normal, and binormal vectors, together with the curvature and torsion, make up the Frenet-Serret apparatus of the curve, and there is a sense in which they say everything there is to say about the curve. I won't go into the details of that, but I would like to point out one important feature of the apparatus.
At each point of the curve, we have three perpendicular vectors, the tangent, the principal normal, and the binormal, each with length 1. We can envision them as defining a coordinate system, with the tangent vector pointing in the x direction, the principal normal in the y direction, and the binormal in the z direction. This system of coordinates turns out to be a very nice system - remember that ellipse! - at least as far as the neighborhood of that one point is concerned. The system changes as we move along the curve, but it does so in a smooth fashion; you can visualize the axes rotating and twisting as you slide along the curve. We call something like this a moving frame.
One last thing: Messrs. Frenet and Serret proved a nice little theorem about their apparatus. We know, from the definitions of the curvature and torsion, what the rates of change of the tangent vector and the binormal are; what about the rate of change of the principal normal? It turns out that the rate of change of the principal normal is the torsion times the binormal, minus the curvature times the tangent. Represented mathematically, this is beautifully symmetric (well, technically, it's antisymmetric). It looks like this:
T'= κN N'=-κT +τB B'= -τN
To verify the middle row, write N'=aT+bN+cB; then a=N'·T, b=N'·N, and c=N'·B. Apply the product rule to differentiate the dot products T·N (which is 0), N·N (which is 1), and B·N (which is 0); all three derivatives must be 0, and you can apply the definitions of the curvature and torsion to finish the job.So that's the story of the Frenet-Serret apparatus. (There are many more things that can be done with it, but I like the imagery of the moving frame sliding along the curve, and the way we can predict the way it changes by using the curvature and torsion.)
For the next post, I'll turn my attention from curves to surfaces.
Addendum: Erk. There are a couple of other details about curves that I should probably talk about, and maybe do an example. Surfaces after that.