Where x and y are differentiable functions about u, then the tangent vector of the curve at a certain point is a vector composed of the first derivative of each component, namely:
Through the above formula, if p '(u) is not 0 at u, then this point becomes the regular point of the curve. A curve whose points are regular everywhere is called a regular curve.
The following formula can be used to calculate the normal vector value of a curve at a certain point:
Through parameter transformation, different parameters can be used to represent the same curve. The differential geometry of curves focuses on arc length, curvature and other properties, which have nothing to do with specific parameters, that is, the values of these properties are equal no matter how the parameters are transformed.
For the above curve, the arc length from the starting point a to any point u on the curve can be expressed as:
In other words, arc length is the integral of tangent vector length to curve parameters. It can be found that the arc length has nothing to do with specific parameters, and the parameter u is mapped from the interval [a, b] to the interval [0, L] (where l is the arc length of the curve).
It can be found that this is a variable upper bound integral function, and the derivatives of both sides are obtained at the same time:
When the modulus of the tangent vector is 1, that is, the tangent vector of the curve is the unit vector field, the parameter u is the arc length parameter of the curve.
Curvature of a curve is the rotation rate of the tangent direction angle of a point on the curve to the arc length, that is, the turning angle of the curve in unit arc length.
Assume that the parameter of the parametric equation of a regular curve is its arc length, and the image is as shown in the above figure. Alpha represents the tangent vector on the curve (that is, the derivative p '(s) of p (s)), and the curvature of the curve is defined as:
Where θ represents the included angle between α(s) and α (s+δ s). To prove this formula, we only need to expand it according to the definition of derivative:
Curvature also has a very important and related property, that is, the radius of curvature, that is, the section of the curve is differentiated as much as possible until it is finally approximated as an arc, and the radius corresponding to this arc is the radius of curvature of that point on the curve. The circle corresponding to this arc is generally called the osculating circle.
We know that the ratio of arc length to radius is radian. For this arc, the curvature value is the ratio of radian to arc length, and the radius value is the ratio of arc length to radian.
For curvature and radius of curvature, the following relationship can be obtained:
Take the expansion of the earth map as an example. The surface of the earth is a closed surface. In order to print a map, it is usually necessary to unfold the surface.
Before expansion, first "cut" along the meridian, and then expand as follows:
It can be found that the north pole is transformed into a line segment AC and the south pole is transformed into a line segment BD.
Suppose that the radius of such a sphere is r, and there are two kinds of coordinates, namely: (x, y, z) and (θ,? )
The former is well understood, that is, the coordinates of a point on a sphere in a three-dimensional Cartesian coordinate system, then the sphere can be expressed by the following implicit equation:
Through this equation, we can quickly judge the positional relationship between a point in space and a sphere.
There are two parameters θ and? Its meaning can be understood by the following figure (much like spherical coordinates)
After understanding, it is not difficult to get two coordinate transformation methods:
Where the value range of θ is [0,2 π],? The value range of is [-0.5π, 0.5π]. With this representation, we can find that the "square" region is mapped to a sphere.
By theta sum? With these two parameters, we can draw two groups of parallel lines similar to latitude and longitude. Through these parallel lines, we can clearly observe the degree of distortion of different parts of the sphere.
Assume that the parametric equation of a three-dimensional surface is as follows
Where x, y and z are differentiable functions about parameters u and v, and ω is the domain of parameters u and v.
Similar to a curve, the measurement of a surface is determined by its first derivative. The partial derivatives of x with respect to parameters u and v are as follows
These two partial derivatives represent tangent vectors on the following two curves.
Obviously, these two equations are surface equations with one parameter fixed and the other parameter parametric.
From the above picture, we can clearly see the specific meanings of c v, c u, x v and X U.
If you want to represent the normal vector of a point on the plane, it is also very simple. The partial derivative of the curve equation about the parameters u and v at a certain point determines two tangent vectors X v, X v, and the normal vector of the surface at this point can be obtained by the cross product of these two vectors.
The above derivative direction is only along the direction of two parameters. If the derivative of a surface in any direction is needed, the concept of directional derivative can be introduced.
It is necessary to give a direction vector when solving the direction derivative. Since the surface equation is given in the form of parametric equation, a direction vector is defined in the parametric space u, v of the curve equation.
Then the surface passes through this point, and the curve equation advancing in the above direction in the parameter space can be expressed as
At this point, the direction derivative of the surface at point (u 0, v 0) in the w direction is:
The vector w is defined in three-dimensional space, and the known direction vector is in two-dimensional parameter space. Now you need to convert it from parameter space to tangent vector on the surface:
This transformation can only be accomplished by applying it to Jacobian matrix:
At this time, the value of Jacobian matrix is:
Through the above process of solving directional derivative, we can find that Jacobian matrix represents a transformation from the domain space of parameters to the coordinate space of surfaces. Through Jacobian matrix, we can know the mapping relationship between these two spaces, such as angle, distance, area and so on.
Suppose there are two unit vectors w 1, w2, and the cosine of the included angle between these two vectors is equal to the inner product of these two vectors.
The expression of vectors in surface space and parameter space is different. One thing that can be made clear about singular numbers is that the included angle between vectors will not change no matter how expressed.
In the above formula, the transposed part of j multiplied by j is called the first basic type of surface.
According to I, the arc length of the following curves should be expressed by parameters:
First, observe the arc length formula of the surface:
Next, the tangent vector w (u t, v t) is expressed by the parameter t, and its module length is:
Finally, the formula of arc length can be obtained by calculation:
Similarly, the area of the surface can be obtained by the following methods:
The definition of surface curvature is extended from the definition of curve curvature. There are countless tangent vectors for a point on a surface. For a point p and a tangent vector t on a surface, curvature can be defined as the curvature of a straight line formed by the intersection of the tangent vector t and the normal vector of the surface at this point and the surface at point p.
Write this curvature as:
Where II is the second basic type.
The above function about surface curvature will have two extreme values (maximum and minimum) when it changes in the tangential direction, which is generally called principal curvature. If the two extreme values are not equal, the two tangent vectors when taking these two extreme values are called principal directions. If the two poles are equal, this point on the surface is called navel, all tangent vectors of this point on the surface can be called principal direction, and the curvature of this point on the surface is equal in all directions. In particular, if and only if the surface is spherical or flat, all points on it are umbilical.
For the two principal curvatures of a surface and their curvatures in any direction at the same point, there is the following relationship:
Where ψ is the included angle between the principal direction t 1 and the specified direction t, it can be seen that the curvature of a surface is only determined by its two principal curvatures, which means that the normal curvature in any direction is a convex combination of these two principal curvatures, and it can also be concluded that the principal directions are always orthogonal to each other.
The properties of a certain area of a surface can also be expressed by a curvature tensor, and the curvature tensor c can be obtained by the following methods:
Where D is a third-order square matrix with diagonal elements κ 1, κ 2,0, and P is also a third-order square matrix, which consists of three column vectors: T 1, T2 and n.
In addition, there are two widely used methods to describe curvature:
Gaussian curvature can divide points on the surface into three categories:
Gaussian curvature and average curvature are usually used for visual analysis of surfaces.
In differential geometry, those properties that only depend on the first basic type are called intrinsic. Intuitively, they can only be derived from the two-dimensional features of the surface. For example, the length and angle of a curve on a surface are inherent.
For Gaussian curvature and average curvature, the former is invariant under equidistant transformation, so it is intrinsic, that is, Gaussian curvature can be directly determined by the first basic form; The latter is not, it depends on the surface.
Inherent is usually used to indicate the independence of parameters.
Generally speaking, the divergence of function gradient is called Laplace operator. For the binary function f (u, v), its second-order difference operator (Laplace operator) in Euclidean space can be written as:
Laplace operator can also be extended to second-order manifold surface S, and its extended form is called Laplace-beltrami operator, which is defined as:
For a specific point x on the surface, the Laplace-beltrami operator and its average curvature have the following relationship:
Although this formula shows that there is a certain relationship between the average curvature (extrinsic) and the Laplace-beltrami operator, the Laplace-beltrami operator itself only depends on the first basic type and is intrinsic.
Because 3D meshes are not continuous, the above discussion is based on smooth surfaces. When the above operators are applied to a 3D mesh, it is necessary to treat the mesh as a very rough surface, and then calculate the differential properties of this approximate surface through the mesh data.
The general idea is to calculate the average value of differential properties of a point in the grid and its adjacent points.
When the area of a point in the grid and its adjacent points is large, the differential properties obtained by calculating the average value will be very stable; When the area is small, subtle changes will be better preserved.
This area is usually defined in the following three ways. The main difference lies in the way that triangles surround vertices:
In the picture on the right, if the triangle is an obtuse triangle, take the midpoint of the opposite side of the center point, otherwise take the outer center of the triangle.
In a 3D grid, it is relatively easy to calculate the normal vector of a triangle, just take two edge vectors and cross multiply them:
If you want to calculate the normal vector of the vertex, you should also consider the weighted average of the normal vectors of adjacent triangles around the vertex:
The following methods are usually used to obtain the weight αT:
Also based on the weighted average method, the coordinates of a point on a triangle in the grid can be weighted average by the gradient of three vertices according to the three weights of the center of gravity coordinates.
For piecewise linear function f, it has a corresponding value at the vertex of the triangle. Lagrange interpolation polynomial can be considered as a function value representing any point on a triangle (u is a two-dimensional parameter):
Because the basis function b of Lagrange interpolation formula has the following properties
Both sides can be obtained simultaneously by gradient operation.
After removing Bi, the original formula is
The gradient of the basis function at vertex I is a vector from vertex I along the opposite height direction, and the modulus length of the vector is the reciprocal of the height. After simplification (the vector is rotated by 90 degrees and divided by the length of the base to get the unit vector, and then divided by the height, where the length of the base multiplied by the height is twice the area), it is:
After substitution, the gradient of piecewise linear function on triangle can be obtained as follows
Intuitively, this form is a vector with the center point I as the starting point and the average value of adjacent vertices as the end point.
Since the average curvature h of the plane is 0, the result of the operator should be 0, but the result of the above formula is not necessarily non-zero, so this method is not suitable for non-equidistant grids. .
This method only considers the connectivity of the grid, so its application scope is limited.
This form is more accurate, directly calculate the average area around the vertex v i (as mentioned above, there are several ways to get it), and then integrate the divergence of its gradient with the surface, and then use the divergence theorem (Gaussian formula) to expand the calculation, and finally get:
Because Laplace operator is defined as the divergence of gradient, given a vector w of each triangle T (such as the gradient vector under a piecewise linear function f), its divergence is
According to the above formula, the average curvature in discrete form can be obtained:
In the article [Meyer et al. 03], the expression of Gaussian curvature in discrete form is mentioned:
According to the relationship between Gaussian curvature, average curvature and two principal curvatures, the calculation method of principal curvatures can be obtained:
Where β( e) represents the dihedral angle of the triangle plane adjacent to the edge E, e ∩A( v) represents the length of the edge E in the area A, and ο represents the unit vector of the edge E. ..