渐屈线

hbghlyj

infinitesimal calculus, 1897, Lamb, 1946 reprinted version, from page 349
142. Evolutes.
The 'evolute' of a curve is the locus of its centre of curvature. Since the centre of curvature is (Art. 133) the intersectoin of two consecutive normals, the evolute is also the envelope of the normals to the given curve. Hence the normals to the original curve are tangents to the evolute*.(footnote * : It being evident that the exceptional cases noted at the end of Art. 141 cannot present themselves in the envelope of a straight line)
Ex. 1. In the parabola $y^2=4ax$,...(1)
we have $x=a\cot^2\psi,$  $y=2a\cot\psi$,...(2)
and (by Art.134, Ex. 4) $\rho=-2a/\sin^3\psi$,...(3)
The coordinates of the centre of curvature are therefore $\left.\begin{array}r\xi=x-\rho\sin\psi=3x+2a,\\\eta=y+\rho\cos\psi=-y^3/4a^2\end{array}\right\}$,...(4)
Hence $\eta^2=y^6/16a^4=4x^3/a=\frac4{27}(\xi-2a)^3a$.
The evolute is therefore the semi-cubical parabola $ay^2=\frac4{27}(x-2a)^3$...(5)
Otherwise: it is shewn in books on Conics that the equation of the normal is of the form $y=m(x-2a)-am^3$...(6)
To find the envelope of this we differentiate partially with respect to $m$, and obtain $x-2a=3am^2,$  $y=2am^3$...(7)
The elimination of $m$ leads again to the result (5).
The curve is shewn in Fig 116.

Ex. 2. The normal at any point of the ellipse
$x=a\cos\phi,$  $y=b\sin\phi$...(8)
is $\frac{ax}{\cos\phi}-\frac{by}{\sin\phi}=a^2-b^2$. ...(9)
Differentiating with respect to $\phi$ we find $\frac{ax}{\cos^3\phi}=-\frac{by}{\sin^3\phi},=\lambda,$ say. ...(10)
Substituting in (9), we have $\lambda=a^2-b^2$...(11)
Hence the coordinates of the centre of curvature are $x=\frac{a^2-b^2}a\cos^3\phi,$  $y=-\frac{a^2-b^2}b\sin^3\phi$; ...(12)
and the evolute is $(ax)^{\frac23}+(by)^{\frac23}=(a^2-b^2)^{\frac23}.$ ...(13)
This curve, which may be obtained by orthogonal projection from the astroid, is shewn in Fig. 117.
The centres of curvature at the points $A,B,A',B'$ are $E,F,E',F'$, respectively.

Ex3. To find the evolute of a cycloid.
At any point $P$ on the cycloid $APD$(Fig. 118), we have, by Art. 134, Ex. 2, $\rho=2PI$. ...(14)
Let the axis $AB$ be produced to $D'$, so that $BD'=AB$; and produce $TI$ to meet a parallel to $BI$, drawn through $D'$, in $I'$. If a circle be described on $II'$ as diameter, and $PI$ be produced to meet its circumference in $P'$, we have $P'I=PI$, so that $P'$ is the centre of curvature of the cycloid at $P$. And since the arc $P'I'$ is equal to the arc $TP$, and therefore to $BI$ or $D'I'$, the locus of $P'$ is evidently the cycloid generated by the circle $IP'I'$, supposed to roll on the under side of $D'I'$, the tracing point starting from $D'$. That is, the evolute is a cycloid equal to the original cycloid, and having a cusp at $D'$.
It appears, further, from Art. 122(4), that the cycloidal arc $P'D$ is equal to $2IP'$, or $P'P$. Hence
$\operatorname{arc}D'P'+P'P=\text{const.}$ ... (15)
The lower cycloid in Fig. 118 is therefore an 'involute' (Art. 144) of the upper one* (footnote *: This example is interesting historically in connection with the theory of the cycloidal pendulum. The results are due to Huyghens (1673).)
Whenever a curve is defined by a relation between $p$ and $\psi$, say
$p=f(\psi)$, ...(16)
the evolute is given by
$p=f'(\psi)$, ...(17)
provided that in (17) the origin of $\psi$ be supposed moved forwards through a right angle. This is seen at once on reference to Fig.108, p.338, since $OU$, the perpendicular from the origin on the tangent to the evolute, is equal to $PZ$, or $\mathrm dp/\mathrm d\psi$, when the symbols refer to the original curve.

Ex. 4. To find the evolute of an epi- or hypo-cycloid.
If in Fig.81, p.297, a perpendicular $p$ be drawn from $O$ to $TP$, the tangent to the epicycloid at $P$, we have
$p=OT\cos PIC=(a+2b)\cos\frac12\phi,$
or $p=(a+2b)\cos\frac a{a+2b}\psi$,...(18)
If the origin of $\psi$ correspond to a cusp instead of to a vertex, the cosine of the angle must be replaced by the sine.
Hence, for the evolute, we have
$p=-a\sin\frac a{a+2b}\psi$,...(19)
which can be brought to the same form as (18) by an adjustment of the origin of $\psi$. The evolute is therefore a similar epicycloid in which the dimensions are reduced in the ratio $a/(a+2b)$.
For a hypocycloid we have merely to change the sign of $b$*.(footnote *: It appears on examination that the equation
$p=c\cos \psi$, or $p=c\sin m\psi$,
represents an epi- or a hypo-cycloid according to $m\lessgtr 1$,provided we include the pericycloids among the epicycloids, in accordance with the definition of Art.123.
The pedal of an epi- or a hypo-cycloid with respect to its centre is therefore an epicyclic of the special type referred to in Art.125, Ex.2. Thus Fig.92 represents the pedal of a four-cusped epicycloid, and Fig.94 that of a four-cusped hypocycloid.)

hbghlyj

Frenet–Serret formulas - Wikipedia
Some Comments on the Derivative of a Vector with applications to angular momentum and curvature (E. L. Lady)(October 18, 2000)
CURVATURE (E. L. Lady)
本楼存放插图及源码
Fig. 116.

1. (Geogebra to Asymptote)
3. import contour; import graph; size(18.60240178339251cm);
4. real labelscalefactor = 0.5;
5. pen dps = linewidth(0.7) + fontsize(30); defaultpen(dps);
6. pen dotstyle = red;
7. real xmin = -0.23682272304871493, xmax = 10, ymin = -6.313126059693872, ymax = 5.339502023723345;
9. real parabola1 (real x) {return x^2/2/2.0;}
10. draw(shift((0.,0.))*rotate(-90.)*graph(parabola1,-16.,16.), linewidth(2.));
11. real implicitf1 (real x, real y) { return 1.1851851851851851+1.0*y^2-1.7777777777777777*x^1+0.8888888888888888*x^2-0.14814814814814814*x^3; }
12. draw(contour(implicitf1, (xmin,ymin), (xmax,ymax), new real[]{0}, 500), linewidth(2.));
13. draw((xmin, -1.3335719977961158*xmin + 5.038787471286494)--(xmax, -1.3335719977961158*xmax + 5.038787471286494), linewidth(2.));
14. draw((xmin, 0.7498657752656905*xmin + 1.3335719977961156)--(xmax, 0.7498657752656905*xmax + 1.3335719977961156), linewidth(2.));
15. draw((xmin, 0)--(xmax, 0), linewidth(2.));
16. dot((0.,0.),dotstyle);
17. label("$A$", (0.07670090251408015,0.16636220193722667), NE * labelscalefactor);
18. dot((1.,0.),linewidth(4.pt) + dotstyle);
19. label("$S$", (1.0695257167962644,0.13152624354136055), NE * labelscalefactor);
20. dot((7.335242819169654,-4.743286949393193),dotstyle);
21. label("$C$", (7.409670144843899,-4.571328139900565), NE * labelscalefactor);
22. dot((1.7784142734655415,2.667143995711924),linewidth(4.pt) + dotstyle);
23. label("$P$", (1.6,2.8138950400230516), NE * labelscalefactor);
24. dot((6.6853758007523085,0.),dotstyle);
25. label("$X$", (6.747786935322442,0.16636220193722667), NE * labelscalefactor);
26. clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);

Copy the Code

Fig. 117.

1. (Geogebra to Asymptote)
3. import contour; import graph; size(5.824883720930265cm);
4. real labelscalefactor = 0.5;
5. pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
6. pen dotstyle =linewidth(3)+red;
7. real xmin = -2.895581395348854, xmax = 2.929302325581411, ymin = -2.424914320685443, ymax = 2.3639228886169095;
9. pair A = (2.5,0.), B = (0.,2.), F = (0.,-1.125);
10. pair f1 (real t) {return (2.25/2.5*(cos(t))^(3),-2.25/2*(sin(t))^(3));}
11. draw(graph(f1,0.,6.283));
12. draw(shift((0.,0.))*rotate(0.)*xscale(2.5)*yscale(2.)*unitcircle, linewidth(2.));
13. draw((-2.5,0.)--A, linewidth(1.));
14. draw(B--(0.,-2.), linewidth(1.));
15. dot(A,dotstyle);
16. draw((0.44220440724556953,-0.26076242681208206)--(1.9727264205498716,1.2285537434663873));
18. label("$A$", (2.57,0.007876376988989481), NE * labelscalefactor);
19. dot((-2.5,0.),dotstyle);
20. label("$A'$", (-2.93,0.06), NE * labelscalefactor);
21. dot(B,dotstyle);
22. label("$B$", (-0.1,2.079969400244815), NE * labelscalefactor);
23. dot((0.,-2.),dotstyle);
24. label("$B'$", (-0.2,-2.4), NE * labelscalefactor);
25. dot((0.9,0.),dotstyle);
26. label("$E$", (0.75,-0.4), NE * labelscalefactor);
27. dot((-0.9,0.),dotstyle);
28. label("$E'$", (-1.0767441860465181,-0.4), NE * labelscalefactor);
29. dot((0.,1.125),dotstyle);
30. label("$F'$", (0.08,0.98), NE * labelscalefactor);
31. dot(F,dotstyle);
32. label("$F$", (0.05,-1.243053855569157), NE * labelscalefactor);
33. dot((0.44220440724556953,-0.26076242681208206),dotstyle);
34. label("$C$", (0.4427906976744206,-0.5370073439412462), NE * labelscalefactor);
35. dot((1.9727264205498716,1.2285537434663873),dotstyle);
36. label("$P$", (2.0467441860465225,1.3125275397796945), NE * labelscalefactor);
37. clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);

Copy the Code

Czhang271828

这是渐屈线的微分几何解释, 可以提供一些思路.

记 $\gamma(t)=(x(t),y(t))$ 表示平面曲线, 则$\gamma'(t)$ 与切向平行. 若对于 $t\in[a,b]$ 均有 $\gamma'(t)\neq 0$, 则称曲线 $\gamma(t)$ 在 $t\in[a,b]$ 时正则. 由于弧长函数
$$
\gamma(t)=\int_a^t|\gamma'(t)|\mathrm d t,\quad t\in[a,b].
$$
是单调非减的, 因此可以记 $\gamma(s)$ 为 $\gamma(t)$ 的弧长参数表示使得 $\left|\dfrac{\mathrm d\gamma}{\mathrm d s}\right|\equiv 1$. 约定弧长参数下 $f'=\dfrac{\mathrm d f}{\mathrm d s}$, 特别地, $|\gamma'|=1$

记 $\vec t=\gamma'$ 为单位切向量, $\vec n= \vec z\times \vec t$ 为单位法向量, 即单位法向量为单位切向量逆时针转 $\pi/2$. 我们注意到 $\left< \vec t',\vec t\right>=\dfrac{1}{2}[|t|^2]'=0$, 从而 $\vec t'$ 与 $\vec n$ 平行. 记 $\vec t'=\kappa \vec n$, 其中 $\kappa$ 为曲率, $|\kappa|={|t'|}$ 为曲率大小 (平面曲线曲率分正负, 因为约定$\vec n= \vec z\times \vec t$), $|t'|$​ 为曲率.

同理, $\vec n'$ 与 $\vec t$ 平行. 注意到
$$
\left< \vec n',\vec t\right>=\left< \vec n,\vec t\right>'-\left< \vec n,\vec t'\right>=0-\kappa.
$$
从而得到活动正交框架 $\{\vec t,\vec n\}$ 满足的微分关系:
$$
\begin{pmatrix}\vec t'\\\vec n'\end{pmatrix}=\begin{pmatrix}0&\kappa\\-\kappa&0\end{pmatrix}\cdot \begin{pmatrix}\vec t\\\vec n\end{pmatrix}.
$$
渐屈线无非"曲率圆"的圆心轨迹
$$
\tilde \gamma=\gamma+\dfrac{1}{\kappa}\vec n.
$$
对于非弧长参数但正则的平面直角坐标系上的曲线 $\gamma(t)=(x(t),y(t))$ 而言,
$$
\dfrac{1}{\kappa}\vec n=\dfrac{1}{\kappa^2}\cdot \vec t'=\dfrac{(x_t^2+y_t^2)^3}{(x_ty_{tt}-x_{tt}y_t)^2}\cdot(x_t,y_t).
$$
所以计算得
$$
(\tilde x(t),\tilde y(t))=\dfrac{(x_t^2+y_t^2)^3}{(x_ty_{tt}-x_{tt}y_t)^2}\cdot(x_t,y_t)+(x,y).
$$
渐屈线上尖点为不可微点, 满足 $|\tilde \gamma'|=0$, 即
$$
\tilde \gamma'=\vec t+\dfrac{1}{\kappa}(-\kappa\vec t)-\dfrac{\kappa'}{\kappa^2}\vec n=-\dfrac{\kappa'}{\kappa^2}\vec n.
$$
从而正则曲线的尖点一定为曲率取极值的点. 反之也成立, 因为 $\vec t$ 与 $\vec n$ 可微, 而渐屈线的切方向在曲率增减性改变时"突然调头".

以上直接采用曲率来定义渐屈线, 当然也可以定义渐屈线为法线的包络线: 根据曲率定义, 这两种定义是同等的.

再比如, 正则平面闭曲线 $\alpha$ 的渐屈线 $\tilde \alpha$ 为分段正则的闭曲线 (不考虑直线与圆弧的情形), 且至少有四个尖点. 因为 $\alpha$ 有唯一的外接圆, 且切点数量至少为 $2$. 显然切点处为曲率极值, 相邻切点间也包含了至少一个极值, 总共至少有 $4$ 个极值, 从而渐屈线尖点数量至少为 $4$.

hbghlyj

Fig. 118.

1. Geogebra to Asymptote
3. import graph; size(3.6cm);
4. pen dps = linewidth(0.7) + fontsize(1); defaultpen(dps);
5. pen dotstyle = linewidth(2) + blue;
6. draw(circle((0.,-1.), 1.));
7. pair f1 (real t) {return (t-sin(t)+pi/2,-1+cos(t));}
8. draw(graph(f1,-3.141592653589793,0.) , green);
9. draw(circle((0.,1.), 1.));
10. draw((-1.5707963267948966,-2.)--(1.5707963267948966,-2.));
11. draw((1.5707963267948966,-2.)--(1.5707963267948966,2.));
12. draw((1.5707963267948966,2.)--(-1.5707963267948966,2.));
13. draw((-1.5707963267948966,2.)--(-1.5707963267948966,-2.));
14. pair f2 (real t) {return (t-sin(t)-pi/2,1+cos(t));}
15. draw(graph(f2,0.,3.141592653589793) , red);
16. draw((1.,-1.)--(-1.,1.));
17. draw((0.,2.)--(0.,-2.));
18. draw((-1.5707963267948966,0.)--(1.5707963267948966,0.));
19. dot((0.,0.),dotstyle);
20. label("$I$", (-0.2,-0.24426664626683414), NE);
21. dot((0.,-2.),dotstyle);
22. label("$T$", (-0.07434232558140494,-2.2667675764994), NE);
23. dot((1.,-1.),dotstyle);
24. label("$P$", (1,-1.1533705997552097), NE);
25. dot((1.5707963267948966,0.),dotstyle);
26. label("$D$", (1.6,-0.0399736230110194), NE);
27. dot((-1.5707963267948966,-2.),dotstyle);
28. label("$A$", (-1.9,-2.2), NE);
29. dot((-1.5707963267948966,0.),dotstyle);
30. label("$B$", (-1.9,-0.060402925336600875), NE);
31. dot((-1.5707963267948966,2.),dotstyle);
32. label("$D'$", (-1.9,2.084673818849454), NE);
33. dot((0.,2.),dotstyle);
34. label("$I'$", (0.038018837209293166,2.084673818849454), NE);
35. dot((-1.,1.),dotstyle);
36. label("$P'$", (-1.4,0.7976277723378211), NE);

Copy the Code

(为了在一个页面内嵌多个svg,必须对id做处理以免重复而use元素引用id时出错,这里把id前面加上字母z)

Czhang271828

hbghlyj 发表于 2022-8-19 12:58
红色部分, 我觉得应该是 $|\kappa|=\color{red}{|\vec t\hspace.7mm'|}$

对, 这是上分句直接推出的, 已改正.

Czhang271828

hbghlyj 发表于 2022-8-19 22:49
还有这里:

Czhang271828

hbghlyj 发表于 2022-8-20 03:55
好像缺少\vec

这个小问题, $Ax=y$ 也没见得写 $\vec x$. 强调大小时统一用模长即可.

青青子衿

hbghlyj 发表于 2022-1-23 07:21
本帖最后由 hbghlyj 于 2022-8-19 06:11 编辑 Frenet–Serret formulas - Wikipedia
Some Comments on the ...

椭球好像也有渐屈曲面
Apollonius Problem and Caustics of an Ellipsoid
researchgate.net/publication/370656616