The curve in (1)R3 is a continuous image of one-dimensional space, so it is one-dimensional.
(2) The curve in r 3 can be obtained by twisting a straight line.
(3) A certain value of a parameter refers to a point on a curve, but the reverse is not necessarily true, because we can consider a self-intersecting curve.
Differential geometry is a subject that uses calculus to study geometry. In order to apply the knowledge of calculus, we can't consider all curves, even continuous curves, because continuity is not necessarily differentiable. This requires us to consider differentiable curves. But differentiable curves are not very good, because there may be some curves whose tangent direction is uncertain at a certain point, which makes it impossible for us to start from the tangent. This requires us to study this kind of curve whose derivative is not zero everywhere, which we call regular curve. Regular curve is the main research object of classical curve theory.
Curve: Any continuous line is called a curve. Include straight lines, broken lines, line segments, arcs, etc. The curve is a 1-2-dimensional graph, referring to the fractal dimension space. Curves that turn around are generally infinite in length and zero in area. At this time, the curve itself is a space larger than 1 smaller than 2 dimensions. One of the main objects of differential geometry research. Intuitively, the curve can be regarded as the trajectory of particle motion in space. A stricter definition of a curve is the mapping of R: α, b) to E3. Sometimes, this mapping image is also called a curve.
Specifically, let Oxyz be Cartesian Cartesian coordinate system in Euclidean space E3, and r be the radius of the point on curve C, so there is. The above formula is called the parametric equation of curve C, t is called the parameter of curve C, and the advancing direction of curve C is naturally determined according to the direction of parameter increase (Figure 1). In curve theory, regular curves are often discussed, that is, curves whose derivatives of three coordinate functions x(t), y(t) and z(t) are continuous and zero for any T. For regular curves, the arc length S can always be regarded as a parameter, which is called natural parameter or arc length parameter. The arc length parameter S is defined by and represents the length of curve C from r(α) to r(t). It is also assumed that the coordinate function of curve C has a third-order continuous derivative, that is, the curve is C3-order. Let the parametric equation of the regular curve C be r=r(s), s be the arc length parameter, and p(s) be a fixed point on the curve C with parameter S, that is, the radial diameter R (s). Q (s+Δ s) is a point near P on C. When Q approaches P along curve C, the limit position of secant pQ is called the tangent of curve C at point P, and the plane passing through point P and perpendicular to the tangent is called the normal plane of curve C at point P. The tangent of curve C at point P and its neighboring point R determine a plane σ. The limit position of σ is called the closed plane of curve C at point P, and its normal at point P is called the subnormal of curve C at point P, and the plane determined by the tangent and subnormal of curve C at point P is called the tangent plane of curve C at point P. The normal at point P is called the main normal of curve C at point P (Figure 2).
curve
The derivative of the arc length parameter S is expressed by the following formula
That and b(s)=t(s)×n(s) are the unit vectors of tangent, principal normal and subnormal of curve C at point p(s), respectively, t(s) points to the positive direction of curve C, and n(s) points to the concave edge of curve C. T(s), n(s) and b(s) form a right-handed system in turn, which are called tangent vector, principal normal vector and secondary normal vector of curve C at point p(s) respectively. {r(s), t(s), n(s), b(s)} is called Freinet frame of curve C at point p(s). curve
Every point in C has a Freinet frame. In order to study the transformation relationship between Freinet frames of two adjacent points on a curve, we should discuss the derivatives of t(s), n(s) and b(s) about S, which can be expressed linearly by frame vectors, which is the basic formula of the following curve theory (Freinet formula):
Where k(s) and τ(s) are respectively called curvature and torsion of curve C at point p(s). bend
bend
This is the included angle between tangent vectors t(s) and t (s+δ s). So curvature measures the rate of change of the included angle of tangent vectors of two adjacent points on the curve relative to the arc length. The curvature of a straight line is always 0. The curvature of a circle is equal to the reciprocal of its radius. When the curvature of curve C at point p(s) is k≠0, take point Q along the positive direction of n(s) on the main normal line of point p(s) so that pQ= 1/k, and the circle centered on 1/k on the closed plane of point P is called the curvature circle or closed circle of curve C at point P, which is close.
turn around
, its absolute value
Measure the change rate of the included angle between the sub-normal vectors of two adjacent points on the curve to the arc length. A plane curve is a curve with constant torsion. If a space curve is not on a plane, it is called a deflection curve.
If the curvature and torsion of point p0(s0) are not zero, then p0 is taken as the origin and the tangent, principal normal and secondary normal of the curve are taken as the coordinate axes. Near p0, the curve can be approximately expressed as:
So the approximate shape of curve c around p0. The arc length s, curvature k(s) and torsion τ(s) of a curve are motion invariants. Conversely, the curvature and torsion of the curve completely determine the shape of the curve. Specifically, if two continuous functions k (s) >: 0 and τ(s), s∈α, b), there are curves with curvature and torsion of k (s) and τ(s) respectively, and these curves can overlap each other through a motion in space. A curve whose torsion is always zero is a plane curve. Let Oxy be the Cartesian coordinate system of Euclidean plane E2, then the parametric equation of plane curve C is r=r(s)=(x(s), y(s)), and s is the arc length parameter, and Freinet formula can be written as follows.
Where nr is the unit normal vector, so the directed angle from t(s) to nr(s) is. Kr(s) is called relative curvature, kr >；; 0 and kr < 0 indicate that the curve turns left and right respectively. spiral
C is a torsion curve. If its curvature and torsion have a fixed ratio, it is called a spiral. Characterized in that the tangent line forms a fixed angle with the fixed direction. In particular, if curvature and torsion are non-zero constants, then c is a cylindrical spiral, that is, it is on the cylindrical surface and forms a fixed angle with the straight generatrix. It is the trajectory of a particle rotating around a straight line (spiral axis) at a uniform speed and moving along the axis at a uniform speed. Bertrand curve
If the deflection curve c satisfies λ k (s)+μ τ; (s)= 1, where λ and μ are constants, λ >; 0, called Bertrand curve. Such a curve can establish a one-to-one correspondence with another curve, so that the main normals of the corresponding points coincide. On the contrary, this property is also a sufficient condition for a curve to become a Bertrand curve. Each of these c's is called the companion of the other. Tangents of two Bertrand couple lines at corresponding points are fixed angles. Tapered and extended lines
If the tangent of curve C 1 is the normal of another curve C2, then C 1 is called the tapered line or involute of C2, and C2 is called the involute or involute of C 1. It can be proved that the tooth profile curve of the gear meshing with the tooth profile curve is also an involute, and usually the tooth profile curve of the gear adopts a circular involute. When all or a section of a curve is taken as the research object, the geometric properties of the whole curve are obtained. Let the parameter equation of curve C be r=r(s), s∈α, b), and s is the arc length parameter. If its starting point and ending point coincide, and r(α)=r(b), then this curve is closed, which is called a closed curve. If its tangent vectors coincide at this point, that is, r┡(α)=r┡(b), and they no longer intersect, it is called a simple closed curve. For a regular closed curve C, the starting point of its tangent vector t(s) is at the origin, and the ending locus of t(s) is the closed curve on the unit sphere, which is called the tangent image or tangent scale of curve C. The length of the tangent image of C is
On the right side of the equation is the integral of the curvature k(s) of the closed curve C along C, which is naturally called the total curvature of the curve C. The total curvature of a regular closed curve is equal to the length of its tangent image. There are two theorems about the total curvature boundary of regular closed curves. Finch theorem
The total curvature of a regular closed curve C, and the equal sign only holds if C is a plane convex closed curve. This theorem gives the lower limit of the total curvature of a regular closed curve, and Bai Zhengguo extended this theorem to a piecewise smooth closed curve. Farley-Milnor theorem
Complete curvature of closed curve in simple regular kink space.
Integral of closed curve c along its own torsion τ(s)
Naturally, it is called the total torsion of c, and the total torsion of the closed curve on the sphere is equal to zero. On the contrary, if the total torsion of any closed curve on a nonplanar surface is equal to zero, then this surface is a sphere or a part of a sphere.
Let C be a plane regular closed curve, then when a point circles C, the tangent image t(s) of the curve C will circle several times on the unit circumference, and the number of circles ir (counterclockwise is positive, clockwise is negative) is called the rotation of C.
The steering index can be calculated as follows:
Where kr(s) is the relative curvature of c, and the tangent rotation theorem shows that the rotation index ir of the plane simple regular closed curve is equal to 1.
A fixed-length chord on the plane is connected end to end to form a simple closed curve, and the plane is divided into two parts by its common boundary. When the area it encloses is the largest, its shape is round. The following conclusion is more accurate: Let curve C be a plane regular simple closed curve with length L, and A be the area enclosed by C, then L2-4A≥0, and the equal sign holds if and only if C is a circle. The above inequalities have been generalized by various methods, and such problems are called isoperimetric problems. For plane curve, similar to the basic theorem of space curve theory, its shape is determined by its relative curvature kr(s), so the extreme value of kr(s) is naturally interesting. The stagnation point of the relative curvature kr(s) is called the vertex of the curve. For convex closed curves, that is, curves located on the tangent side of each point, the famous four-vertex theorem holds: a plane convex closed curve has at least four vertices, because an ellipse has only four vertices, this conclusion cannot be improved. In addition, Cauchy-Croft formula can be used to calculate the length of plane regular curves (see integral geometry).