Due to the limitation of the author's level, please correct me if there are any mistakes or explanations in the article.
This paper mainly discusses the definition of differentiable parametric curve, curvature of curve, disturbance rate of curve, definition of regular surface, first basic form and second basic form.
In practice, most of the curves we have to deal with are parametric curves, and they are all differentiable. Therefore, it is better to define a parametric curve than a differentiable parametric curve. The following is the definition of differentiable parametric curve:
As can be seen from the definition, the curve is essentially a corresponding relationship, which can be regarded as a function that maps elements in one-dimensional space to three-dimensional space. So what is one dimension in life? After thinking about it, only one thing is one-dimensional, and that is time! So the parametric curve can also be regarded as the trajectory of a point moving with time in three-dimensional space!
Let's look at an example of a parametric curve-helix.
The tangent of a curve at a certain point is its moving direction and speed at the current point. Tangent vectors are very important for studying curves. Almost all theorems or formulas have tangent vectors, so it is the second most important concept after curve definition.
The tangent vector is defined as:
This is easy to understand. When learning calculus, how to understand that the derivative is instantaneous speed (although instantaneous speed does not exist in real life), in a very short period of time, the ratio of the distance traveled by a car to this period of time is continuously shortened, and the limit is taken to become instantaneous speed. Applying this concept to a curve, the tangent vector of the curve is derived in this way (the tangent vector of point T in the above picture).
The tangent vector of the curve is needed to calculate the arc length of the curve. If the curve has no tangent vector at a certain point, we can't deal with this curve, but the previously defined differentiable parametric curve solves this problem, and the differentiable parametric curve ensures that the curve is differentiable everywhere (with tangent vector).
But it's not enough. We also need a condition that the first derivative of the curve is not 0. This leads to the definition of regular curve.
Then, the calculation formula of arc length is as follows:
It is not used here because there is only one variable in this formula, that is, it is not the upper limit of integral, and other variables, such as this formula, are not variables. So it is essentially a function about the upper limit of integral.
We might as well look at it this way, so that the right side of the integral equation becomes, and after integration, it becomes, so the arc length calculation formula becomes:
That is to say, because it is a constant, the whole formula is a function of.
One more thing to note is that after integration, it is irrelevant, that is, it cannot be expressed as, the length taken will turn the expression into a completely different function, and it is irrelevant anymore.
For a curve, we will naturally consider how "curved" the curve is. So, in mathematics, how to express this degree of bending? The answer is: curvature.
The following is the definition of curvature:
At first glance, the definition of curvature is too harsh, because it requires the parameter of a curve to be exactly its arc length, which seems difficult to achieve, but it is relatively easy to do in our practical application, because:
That is to say, if I have a curve and say yes, it is not the arc length. I can find a curve so that it is exactly the arc length. Besides, this sum is the same curve! Then, the definition of curvature can be applied in the world, because it is the same curve, so the curvature of new is also the curvature of new.
Another advantage of curves with arc length as parameter is that sum is perpendicular. Because if you use the arc length formula, the result is that this is exactly equal to 1, which is always equal to 1. that is
Established. Then we can get the derivatives on both sides of the equation (using the product formula) and simplify them.
In other words, sum is vertical.
Now we know how "curved" the curve is, and its degree of curvature is that it is a scalar. So, what is its geometric meaning?
We can intuitively think that the larger the curve, the more curved it will be near the point. So the geometric meaning of curvature is how much the curve increases in the direction perpendicular to (that is, the direction of) when a small amount is added. This is consistent with our concept of curvature!
For curves, curvature is the degree of bending along the direction, forming a plane with the vertical, that is, with * * *. Curvature is the degree of change of a curve in this plane.
Then, if my curve is a three-dimensional curve, I'm afraid it's not enough to have curvature. Yes, this leads to another concept: disturbance rate.
The following is the calculation process of disturbance rate:
This is a complicated reasoning process, much more complicated than that. The geometric meaning of the disturbance rate is the degree of "bending" of the curve in the direction perpendicular to the plane formed by it, that is, the change of the third dimension in space.
Similarly, from the point of view of practical application, finding regular surfaces is more meaningful than finding surfaces. Therefore, a regular surface is defined as follows. To be honest, the definition of a regular surface is much more complicated than that of a curve, but don't you understand the complexity?
A regular surface is defined as follows:
Let's take a closer look at the definition of regular surfaces. First of all, the neighborhood of a point on a surface is a sphere, and the intersection of its sum is a subset of it (note that it cannot be considered as a surface yet, so the intersection cannot be considered as a patch). Then, there is a one-to-one correspondence between this subset and. Finally, this correspondence also needs to meet the conditions of 1, 2, 3. After that, it can be called a regular surface.
To define the first basic form, we must first define the concept of tangent space of a surface.
Tangent space is defined as follows:
With the definition of tangent space, we can define the first basic form of surface:
The above definition is not easy to understand, let's give an example to illustrate:
Suppose we have a regular surface with parametric curves. The first basic form of this point is:
Among them,
Therefore, the first basic form of the point is: usually, we call this kind of thing the first basic form, but the coefficient of the first basic form!
Note: although there are some materials that say it is the first basic form of surface, it is an incorrect name. There is no first basic form of a surface at all, but only the first basic form of a point on the surface, because we see that this formula calculates a scalar, not a point on the surface at all!
The function of the first basic form is to calculate the length of the curve on the surface, the included angle of the tangent vector of the intersection of two curves, or the area of the surface patch.
Like the first basic form, the second basic form is not curved, but a little on the surface. The geometric meaning of the second basic form is the curvature of the surface at the current point, that is, how "curved" the surface is at that point.
The calculation method of the second basic form is as follows:
As can be seen from the formula, this thing is just a scalar. So I don't like to call it the second basic form of surface, because it will cause great ambiguity, as if it were an expression of surface.
The second basic form of thinking is this. When we studied the curve in front, we calculated the curvature of the curve, and expressed how "curved" the curve is with a quantity. Naturally, we also want to know how "curved" a surface is, which leads to the second basic form.
The second basic form is the distance between two points on the surface in the normal vector direction, as shown in the following figure:
Another representation of the second basic form can be said to be coincidence, because it has no geometric meaning. Let's see:
Because, by taking derivatives on both sides of this formula, we can get
The derivatives of both sides can be obtained.
this means
in other words
That's right. What a coincidence. in other words
It is also the second basic form.
Why is it meaningless? Because dx can be regarded as the tangent vector of X and dN as the tangent vector of N (if N is regarded as the surface of a unit sphere, it can also be said to be a Gaussian mapping, which is of course an inappropriate definition), then the dot product of the two tangent vectors also needs opposite numbers, which is meaningless. Of course, this is just my personal understanding. If readers have a better understanding, please leave a message and tell me. Thank you!
To be honest, these definitions are very basic, but it is not easy to understand these things. Some of them can be expanded into more useful concepts, so I won't sort them out here, because as long as these basic concepts exist, they are not empty!
Differential geometry of curves and surfaces
Elementary differential geometry, second edition, by Andrew Presley.