tangent plane (elementary) (original) (raw)

1 Defining formulae

Let S be a surface in three-dimensional space ℝ3 and let 𝐩=(px,py,pz) be a point of S. Three common methods of describing a surface are as the graph of a function, by an implicit equation, and parametrically. We shall now present formulae for computing the tangent plane in each of these cases.

If the surface is described by an equation of the form

then the tangent plane to S at 𝐩 is described by the equation

| z=pz+∂⁡z∂⁡x|x=pxy=pyz=pz⁢(x-px)+∂⁡z∂⁡y|x=pxy=pyz=pz⁢(y-py). | 1 | | --------------------------------------------------------------------- | - |

More generally, if the surface is described by an equation f⁢(x,y,z)=0, then the equation for the tangent plane is

2 Examples

To illustrate and explain the definition, let us consider three examples.

2.1 Example 1

Let S be the cone specified by the equation

and let 𝐩=(1,0,1). Then we compute the tangent plane as follows:

Evaluating at 𝐩, we have

so the tangent plane is specified by the equation

which simplifies to

Before leaving this example, there are two features which are worth noticing. First, note that the tangent plane intersects the cone not only in the point p, but in the whole line {(s,0,s)∣s∈𝐑}. While it is more typical for the tangent plane to intersect in a point, it is possible for the intersection to be a line or even a curve. In fact, it can be shown that if a surface contains a line, then the intersection of the tangent plane to the surface through a point on the line with the surface will include the line.

Second, note that the tangent plane is not defined at the point (0,0,0). If we try to evaluate the partial derivatives at this point, we run into the indeterminate form 0/0. Sometimes, one can make sense of such a form by using L’Hospital’s rule, but in this case, that does not work — the derivatives are simply not defined for the values x=0 and y=0. When it happens that it is not possible to define a tangent plane at a certain point, that point is known as a singular point of the surface. In our case, (0,0,0) is a singular point of the cone. Looking at the surface, it is not hard to see why this point is singular — this point is the vertex of the cone.

2.2 Example 2

To illustrate the computation of tangent planes to a surface which is described by an implicit equation, we shall now compute the tangent plane to the sphere of radius R centred about the origin. The sphere may be described by the equation

In the notation of the last section, we have

Computing partial derivatives, we find that

If 𝐩 is a point which lies on the sphere, the equation of the tangent plane through 𝐩 is as follows:

We may simplify this equation by cancelling the factors of 2 and using the fact that px2+py2+pz2=R2:

Having derived this equation, let us note an interesting fact about the tangent plane to the sphere. Let L be the radius through 𝐩. (By definition, L is the line connecting 𝐩 with the point (0,0,0).) It is easy to see that L is perpendicular to the tangent plane. This is a generalization to three dimensions of the well-known fact of plane geometry that the tangent to a circle through a point on the circle is perpendicular to the radius through that point.

2.3 Example 3

To gain some practise with computing tangent planes to parameterized surfaces, let us compute a tangent plane to a sphere a different way. A sphere of radius R centred about the origin may be also described by the parametric equations

Let us compute the tangent plane through the point 𝐩=(0,R,0) using this description of the sphere. First, we notice that to describe the point 𝐩, we should take u0=π/2 and v0=0.

Taking derivatives, we find that

Substituting u0 for u and v0 for v, we find

Hence the parametric equations of the tangent plane are

It is clear that this agrees with the answer obtained in the last example.

3 Consistency of definitions

Since we have presented three different formulas for computing the tangent plane, logical consistency demands that we make sure that they indeed define the same plane. Moreover, we should check that if we compute the tangent plane to the same point to the ame surface using two different parameterizations, we obtain the same plane both ways. As we will soon see, consistency follows readily from the chain rule and implicit differentiation.

Since it is rather trivial, let us first derive equations (1) as a special case of (2). Suppose that our surface is given by an equation

Let us define f⁢(x,y,z)=g⁢(x,y)-z. Our surface is described by the equation f⁢(x,y,z)=0. By formula (2), its tangent plane at a point 𝐩 is described by the equation

Making minor rearrangements and recognizing that ∂⁡g/∂⁡x and ∂⁡g/∂⁡y here denote the same entites as ∂⁡z/∂⁡x and ∂⁡z/∂⁡y did in the statement of equation (2) because there z is assumed to equal g⁢(x,y), we see that (2) is indeed a consequence of (1).

To show that equations (2) and (3) are consistent, we will make use of the chain rule. If the same surface is described by a parameterization

and an implicit equation

then we must have

Differentiating with respect to u and using the chain rule, we find that

Likewise, diferentiating with respect to v, we find that

Note that the two equations above are exactly the conditions which are needed to guarantee that (2) and (3) describe the same plane.

Finally, we consider the effect of changing the parameters from u,v to two new parameters u′,v′. By the chain rule, we have

and

and likewise for y and z. Substituting these expressions into formula (3), we obtain

If we define

and

then our system of equations may be rewritten as

which is formula (3) evaluated wit respect to the primed coordinates.

4 Motivation and derivation

So far, we presented formulae for computing the tangent plane. We showed how to use them with examples and verified that they were consistent. However, the defining formulae might as well have been “plucked from out of a hat” since we made no attempt to explain what the formulae mean intuitively of to derive them from more basic principles. The purpose of this section is to make up for this shortcoming by discussing the concept of tangent plane intuitively and proving some results which shed light on this concept.

Just as the tangent to a line may be understood as the limit of a line connecting two nearby points in the limit where the points coalesce, so too the tangent plane can be regarded as a limit. Whereas a line is determined by two points, a plane is determined by three points. Thus, we may try to define the tangent plane as the limit of the plane that passes through three points as the three points approach each other.

Whilst this line of reasoning is correct, a bit of care is needed in order to make it work properly. The problem is that three distinct points do not always determine a unique plane — if the points lie on a straight line, then any plane passing through the line will contain all three points. Furthermore, should the points even be close to collinear, then the position of the plane they span will still depend sensitively on the position of the points. Hence, to guarantee that the limit exists, we need to impose a condition which keeps the points from being or tending to become collinear in the limit. One possible way out of this problem is to insist that the angles formed by the triangle of points stay bounded away from zero. If we make such a stipulation, then we can obtain the tangent plane as a limit.

Theorem Suppose that S is a smooth surface and that 𝐚n, 𝐛n, and 𝐜n are three sequences of points such that

1. 1. For every n, the points 𝐚n, 𝐛n, and 𝐜n are distinct.
2. 1. There exists a point 𝐩∈S such that
      | limn→∞⁡|𝐚n-𝐩|=0limn→∞⁡|𝐛n-𝐩|=0limn→∞⁡|𝐜n-𝐩|=0. |
      | ----------------------------------------------------------- |
3. 1. There exists a constant 0<α<π such that for every n, the angles of the triangle 𝐚n⁢𝐛n⁢𝐜n lie in the interval [α,π-α].

Then the planes spanned by the triplets (𝐚n,𝐛n,𝐜n) tend towards the tangent plane of S through 𝐩 in the limit n→∞

Also, there is a simple relation of tangent planes to tangent lines — the tangent plane of smooth surface S at a point 𝐩 is exactly the union of the tangents to the all the smooth curves which lie on S and pass through 𝐩.

Finally, equation (1) bears a strong resemblance to the multivariate Taylor’s expansion. A weak form of the bivariate Taylor’s theorem with remainder reads

| z-pz=∂⁡z∂⁡x|x=pxy=pyz=pz⁢(x-px)+∂⁡z∂⁡y|x=pxy=pyz=pz⁢(y-py)+c1⁢(x,y)⁢(x-px)2+c2⁢(x,y)⁢(x-px)⁢(y-py)+c3⁢(x,y)⁢(y-py)2 | | ---------------------------------------------------------------------------------------------------------------------------------- |

where the functions c1, c2, and c3 are bounded in a neighborhood of 𝐩. (A strong version of the theorem would be more explicit about the form of these functions. Since we do not need precise information about these functions, the weak theorem suffices.) As x gets closer to to px and y gets closer to py, the remainder terms become increasingly smaller in comparison to the first order terms in the expansion. Therefore, if we are only interested in small values of x-px and y-py it is safe to approximate z by ignoring the remainder. By doing so, we obtain the equation of the tangent plane at 𝐩. In other words, if we are only interested in a small enough neighborhood of the point 𝐩, we may approximate the surface by its tangent plane.

This observation allows one to understand the tangent plane intuitively as a planar approximation to a surface in the neighborhood of a point of the surface. At least in one instance, we make use of this fact in everyday life. With the exception of a few benighted souls who belong to the Flat Earth Society, we know that the Earth is a sphere, yet we find that planar maps afford a perfectly adequate representation of the Earth’s surface for planning excursions within a city. The reason for this is that the size of a city is small compared to the size of the Earth. Hence, the error committed by approximating the spherical Earth on such scales by a tangent plane is a fraction of a percent and it is perfectly reasonable to use a planar map. However, when we move the scale of, say, continents, the errors become proportionately larger and it is not possible to represent such a large portion of the Earth on a planar map without introducing significant distortions.

5 Higher dimensions

The formalism presented here is capable of generalization to more than three dimensions. Since most of the differences between the case of three-dimensions which we have considered and higher dimensions are a matter of bookkeeping to keep track of more variables, we are content to simply present the results. How to use these results and the justifications for them are substantially the same as in three dimensions, so there is no reason to repeat them here with essentially cosmetic changes.

Suppose that S is an m-dimensional surface located in ℝn and let 𝐩 be a point of S. Then we may generalize formulas (1) - (3) as follows:

If the surface S is described by the system of equations

then its tangent space is described by the system

| xn=∑i=1m∂⁡g1∂⁡xi|𝐩⁢xi⋮xn-m+1=∑i=1m∂⁡gn-m+1∂⁡xi|𝐩⁢xi, | 1^{\prime} | | ------------------------------------------------------------- | ----------- |

If the surface S is described by the system of equations

then the tangent space is described by the system

| ∑i=1m∂⁡f1∂⁡xi|𝐩⁢xn=0⋮∑i=1n∂⁡fn-m+1∂⁡xi|𝐩⁢xi=0, | 2^{\prime} | | ------------------------------------------------------ | ----------- |

If the surface is described parametrically on terms of a set of parameters (u1,u2,…,um), then the tangent space may be parameterized as follows:

6 Generalizations

In closing, it might be worthwhile to say a few words about how the notion of tangent space has been generalized by modern mathematicians, even though the methods used lie well beyond the scope of this entry.

There is the intrinsic definition of tangent space. Whereas we considered our surface as situated in an ambient space and defined the tangent space as a certain linear subspace of this space, in the intrinsic method one only considers geometric constructions defined on the surface without reference to an ambient space. This is useful because oftentimes one is interested only in the surface and the ambient space is irrelevant; furthermore, it may be possible to embed the same surface several different ways in an ambient space or describe it in a fashion that makes no reference to an ambient space so it is preferrable not to embed the sufrace in an arbitrary ambient space in order to define its tangent spaces. For this reason, modern differential geometry almost exclusively uses the intrinsic definition; for more on this definition, see http://planetmath.org/node/902this entry and http://planetmath.org/node/2007this entry.