Tangents to a curve as functions of different variables

[etotheipi]

Let's say we have a curve in 2D space that we can represent in both cartesian and polar coordinates, i.e. ##y = y(x)## and ##r = r(\theta)##. If you want the tangent at any point ##(x,y) = (a,b)## on the curve you can just do the first order Taylor expansion at that point $$y(x) = y'(a)x + (b-ay'(a))$$and you get a straight line of gradient equal to that of the curve at the point. But you could also do the same thing with ##r## and ##\theta##, if I just let ##\theta_0## to be the value of ##\theta## at the point ##(x,y) = (a,b)##, i.e. $$r(\theta) = r'(\theta_0)\theta + (r(\theta_0)-\theta_0 r'(\theta_0))$$ This would also be a straight line just touching the curve if it were drawn on an ##(\theta, r)## graph, but on an ##(x,y)## graph it's going to be a weird curve of some variety (Archimedean spiral?). With that said, is the second curve still classed as a tangent to the original curve at the point? You could do lots other things as well, like having ##y## as a function of ##\theta##, or ##x## as a function of ##r##, and come up with lots of different "tangents", though they wouldn't necessarily be straight lines.

So I wondered whether a tangent is a first order representation of the function whose value is identical only at the specified point, in which case all of those examples would be "tangents", or whether only the ##y(x) = y'(a)x + (b-ay'(a))## would be a tangent? And then even if the former is true, are the other forms ever useful? Thank you!

 

[andrewkirk]

When we look at the use of the word 'tangent' in the more generalised setting of calculus of manifolds, we see that it operates as an adjective rather than a noun. The key concept is that of a tangent space at a point P on the manifold, which is the set of all vectors that are in a sense tangent to the manifold at the point - we can call them tangent vectors. Note that it is vectors, not curves that are the members of the space. Those tangent vectors correspond to the gradients in your example.

The next step gets interesting. There are a number of possible definitions of what a vector in the tangent space is. A common definition, and the one I find most intuitive, is that a tangent vector is an equivalence class of all curves that pass through the point P, with the same instantaneous velocity at that point.

Taking that back to your example, we see that both the straight line and the spiral are tangent to the curve at the relevant point. They can have the same velocity, and hence can be associated with the same tangent vector.

This is consistent with the words we use in 2D geometry, where we say that one curve is tangent to another at a point if they touch at that point and do not cross. Both your straight line and the spiral are tangent to the graph at the relevant point.

Note that the definition of the concept of 'tangent to a curve' is purely geometric. It is not affected by the choice of coordinate system - eg Cartesian or Polar.

The use of tangent as a noun is generally confined to Euclidean geometry, usually to talk about tangents to circles. In that context it means a straight line that is tangent to the curve at a point. With that use, it would not include your spiral, even though the spiral is tangent to the curve at the relevant point. That is, it satisfies the criteria for use as an adjective, but not as a noun.

 

[mathman]

Tangent is basically a function defined at a point on the curve. Usually it is represented by a straight line through the point at at an angle defined by the function.
This approach assumes Cartesian coordinates. If you use another coordinate system, you are enterig undefined territory.

 

[etotheipi]

Goodness, thanks a bunch for the explanation! I'd heard of the term before but never really understood what it was about! I've so many questions!

When you construct the tangent space at a chosen point, in 2-space or 3-space, is it similar to to use of intrinsic (n/t) coordinates? As in, you define a normal, tangential and binormal direction depending on the instantaneous focus of the curve at that point? I wonder if this is similar to the instantaneous velocity you mentioned, would that be the tangential component? With the normal component related to the radius of curvature?

Also, more generally, is a "curve" an object that is independent of the variables you choose for your coordinate system when you graph it? As in, it seems to me that ##y = y(x)## and ##r=r(\theta)## in this case represent the same curve, and by choosing the axes to be ##(x,y)##, ##(r, \theta)##, ##(x, \theta)## or whatever else you'd just be drawing the same curve in a different set of coordinates. Evidently it would look different, but would it still be the same object?

Since in that case it seems we could hypothetically define a set of functions which are tangents to the curve, and whether or not they're straight or not would just depend on the coordinate system we choose to look at it in?

Sorry if that doesn't make any sense, and also apologies for all of these questions! I might try and get hold of a textbook somehow to try and have a look at this stuff in more detail. Thanks again!

 

[andrewkirk]

is a "curve" an object that is independent of the variables you choose for your coordinate system when you graph it?

A path is, loosely speaking, a continuous line, which may be wiggly or straight, drawn on a surface. It is a geometric object that is independent of any coordinates that are used to identify points on the surface. In the above case, the surface is an infinite plane - 'the number plane'.

A curve is a path together with a parametrisation, which is a continuous map from the real numbers to the path. The extra information in a curve that is not in the path tells us the velocity at which the path is traversed. There are infinitely many different curves that use a given path, just as many vehicles may traverse a road at different speeds and at different times.

More generally, curves and paths can exist in higher dimensional spaces than just surfaces. For instance the curve through 3D space that is mapped out by the centre of a fired cannonball.

Tangent spaces are not very interesting in the Euclidean case, which is what we have here. The tangent space at a point P on the number plane is just a 2D vector space that can be thought of as the set of all vectors 'in' the number plane that have their tail at P.

Tangent spaces become useful, and more interesting, when dealing with curved spaces, such as the surface of a sphere. The tangent space to a sphere at point P can be thought of as the plane L that touches the sphere at P or, with a little more detail, as the set of all vectors in L whose tails lie at P. In a sense it's the set of all "directions in the plane pointing away from P", except that the tangent vectors differ by magnitude (speed) as well as direction.

 

[etotheipi]

Awesome, thanks for clarifying. I haven't studied curved spaces at all yet so I think it's best I do a bit of reading up to try and get a better hold on some of these things. There's a lot to deconstruct! Thanks a bunch for your help!