Sheng Li (gmachine1729) wrote,
Sheng Li
gmachine1729

On the tangent line and osculating plane of a curve

Originally published at 狗和留美者不得入内. You can comment here or there.

Here, we will be working in [公式] .

Analytic geometry prerequisites

Proposition 1 The distance between a point [公式] and the plane given by [公式] is [公式] .

Proof: A normal vector of the plane is [公式] . We plug in [公式] to get

[公式] the solution of which is [公式] . Since every unit of [公式] corresponds to [公式] of distance, we have for our answer[公式]. [公式]

Proposition 2 The distance between a point [公式] and a straight line given by [公式] can be obtained by the magnitude of a cross product.

Proof: As for this distance, it is obtained by taking the perpendicular with respect the straight line that contains [公式] , which we shall call [公式] . We use [公式] to denote the distance between [公式] and [公式] . One notices that [公式] is equal to [公式] , where [公式] is the angle between the straight line given in the proposition and the straight line connecting [公式] and [公式] . We know that the magnitude of the cross product of two vectors is the product of their magnitudes and the sign of the angle between the two vectors, which completes our proof. [公式]

Preliminary definitions

Definition 1 A regular curve is a connected subset [公式] of [公式] homeomorphic to some [公式] that is a line segment [公式] or a circle of radius [公式] . If the homeomorphism [公式] is in [公式] for [公式] and the rank of [公式] is maximal (equal to 1), then we say this curve is k-fold continuously differentiable. For [公式] , we say that [公式] is smooth.

Definition 2 Let a smooth curve [公式] be given by the parametric equations

[公式] The velocity vector of [公式] at [公式] is the derivative

[公式]The velocity vector field is the vector function [公式] . The speed of [公式] at [公式] is the length [公式] of the velocity vector.

Definition 3 The tangent line to a smooth curve [公式] at the point [公式] is the straight line through the point [公式] in the direction of the velocity vector [公式] .

Tangent line and osculating plane of a curve

We let [公式] denote the length of a chord of a curve joining the points [公式] and [公式] and [公式] denote the length of a perpendicular dropped from [公式] onto the tangent line to [公式] at the point [公式] .

Lemma 1 Let [公式] be continuous in [公式] . Then,

[公式]

Proof: Trivial and left to the reader. [公式]

Theorem 1

[公式]

Proof: We have that [公式] and by Proposition 2 that

[公式]

We have, using properties of limits and keeping Lemma 1 in mind in the process,

[公式]

[公式]

Definition 4 A plane [公式] is called an osculating plane to a curve [公式] at a point [公式] if

[公式]

Theorem 2 At each point [公式] of a regular curve [公式] of class [公式] where [公式] , there is an osculating plane [公式] , and the vectors [公式] are orthogonal to its unit normal vector [公式] .

Proof: Based on the following diagram from [1],

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<p><small>Originally published at <a href="https://gmachine1729.wpcomstaging.com/2021/06/06/on-the-tangent-line-and-osculating-plane-of-a-curve/">狗和留美者不得入内</a>. You can comment here or <a href="https://gmachine1729.wpcomstaging.com/2021/06/06/on-the-tangent-line-and-osculating-plane-of-a-curve/#comments">there</a>.</small></p><div class="RichText ztext Post-RichText"> <p>Here, we will be working in <img src="https://www.zhihu.com/equation?tex=%5Cmathbb%7BR%7D%5E3" alt="[公式]" data-formula="\mathbb{R}^3" /> .</p> <h3>Analytic geometry prerequisites</h3> <p><b>Proposition 1</b> The distance between a point <img src="https://www.zhihu.com/equation?tex=%28x_1%2Cy_1%2Cz_1%29" alt="[公式]" data-formula="(x_1,y_1,z_1)" /> and the plane given by <img src="https://www.zhihu.com/equation?tex=ax%2Bby%2Bcz+%3D+d" alt="[公式]" data-formula="ax+by+cz = d" /> is <img src="https://www.zhihu.com/equation?tex=%5Cfrac%7Bd-%28ax_1%2Bby_1%2Bcz_1%29%7D%7B%5Csqrt%7Ba%5E2%2Bb%5E2%2Bc%5E2%7D%7D" alt="[公式]" data-formula="\frac{d-(ax_1+by_1+cz_1)}{\sqrt{a^2+b^2+c^2}}" /> .</p> <p><i>Proof</i>: A normal vector of the plane is <img src="https://www.zhihu.com/equation?tex=%28a%2Cb%2Cc%29" alt="[公式]" data-formula="(a,b,c)" /> . We plug in <img src="https://www.zhihu.com/equation?tex=%28x_1%2Cy_1%2Cz_1%29+%2B+t%28a%2Cb%2Cc%29" alt="[公式]" data-formula="(x_1,y_1,z_1) + t(a,b,c)" /> to get</p> <p><img src="https://www.zhihu.com/equation?tex=a%28x_1%2Bat%29%2Bb%28y_1%2Bbt%29%2Bc%28z_1%2Bct%29+%3D+d%2C%5C%5C" alt="[公式]" data-formula="a(x_1+at)+b(y_1+bt)+c(z_1+ct) = d,\\" /> the solution of which is <img src="https://www.zhihu.com/equation?tex=t+%3D+%5Cfrac%7Bd-%28ax_1%2Bby_1%2Bcz_1%29%7D%7Ba%5E2%2Bb%5E2%2Bc%5E2%7D" alt="[公式]" data-formula="t = \frac{d-(ax_1+by_1+cz_1)}{a^2+b^2+c^2}" /> . Since every unit of <img src="https://www.zhihu.com/equation?tex=t" alt="[公式]" data-formula="t" /> corresponds to <img src="https://www.zhihu.com/equation?tex=%5Csqrt%7Ba%5E2%2Bb%5E2%2Bc%5E2%7D" alt="[公式]" data-formula="\sqrt{a^2+b^2+c^2}" /> of distance, we have for our answer<img src="https://www.zhihu.com/equation?tex=%5Cfrac%7Bd-%28ax_1%2Bby_1%2Bcz_1%29%7D%7B%5Csqrt%7Ba%5E2%2Bb%5E2%2Bc%5E2%7D%7D" alt="[公式]" data-formula="\frac{d-(ax_1+by_1+cz_1)}{\sqrt{a^2+b^2+c^2}}" />. <img src="https://www.zhihu.com/equation?tex=%5Csquare" alt="[公式]" data-formula="\square" /></p> <p><b>Proposition 2</b> The distance between a point <img src="https://www.zhihu.com/equation?tex=%28x_1%2Cy_1%2Cz_1%29" alt="[公式]" data-formula="(x_1,y_1,z_1)" /> and a straight line given by <img src="https://www.zhihu.com/equation?tex=%28x_0%2Bat%2C+y_0%2Bbt%2C+z_0%2Bct%29" alt="[公式]" data-formula="(x_0+at, y_0+bt, z_0+ct)" /> can be obtained by the magnitude of a cross product.</p> <p><i>Proof</i>: As for this distance, it is obtained by taking the perpendicular with respect the straight line that contains <img src="https://www.zhihu.com/equation?tex=%28x_1%2Cy_1%2Cz_1%29" alt="[公式]" data-formula="(x_1,y_1,z_1)" /> , which we shall call <img src="https://www.zhihu.com/equation?tex=h" alt="[公式]" data-formula="h" /> . We use <img src="https://www.zhihu.com/equation?tex=d" alt="[公式]" data-formula="d" /> to denote the distance between <img src="https://www.zhihu.com/equation?tex=%28x_0%2Cy_0%2Cz_0%29" alt="[公式]" data-formula="(x_0,y_0,z_0)" /> and <img src="https://www.zhihu.com/equation?tex=%28x_1%2Cy_1%2Cz_1%29" alt="[公式]" data-formula="(x_1,y_1,z_1)" /> . One notices that <img src="https://www.zhihu.com/equation?tex=h" alt="[公式]" data-formula="h" /> is equal to <img src="https://www.zhihu.com/equation?tex=d+%5Csin+%5Ctheta" alt="[公式]" data-formula="d \sin \theta" /> , where <img src="https://www.zhihu.com/equation?tex=%5Ctheta" alt="[公式]" data-formula="\theta" /> is the angle between the straight line given in the proposition and the straight line connecting <img src="https://www.zhihu.com/equation?tex=%28x_0%2Cy_0%2Cz_0%29" alt="[公式]" data-formula="(x_0,y_0,z_0)" /> and <img src="https://www.zhihu.com/equation?tex=%28x_1%2Cy_1%2Cz_1%29" alt="[公式]" data-formula="(x_1,y_1,z_1)" /> . We know that the magnitude of the cross product of two vectors is the product of their magnitudes and the sign of the angle between the two vectors, which completes our proof. <img src="https://www.zhihu.com/equation?tex=%5Csquare" alt="[公式]" data-formula="\square" /></p> <h2>Preliminary definitions</h2> <p><b>Definition 1</b> A <i>regular curve</i> is a connected subset <img src="https://www.zhihu.com/equation?tex=%5Cgamma" alt="[公式]" data-formula="\gamma" /> of <img src="https://www.zhihu.com/equation?tex=%5Cmathbb%7BR%7D%5E3" alt="[公式]" data-formula="\mathbb{R}^3" /> homeomorphic to some <img src="https://www.zhihu.com/equation?tex=G" alt="[公式]" data-formula="G" /> that is a line segment <img src="https://www.zhihu.com/equation?tex=%5Ba%2Cb%5D" alt="[公式]" data-formula="[a,b]" /> or a circle of radius <img src="https://www.zhihu.com/equation?tex=1" alt="[公式]" data-formula="1" /> . If the homeomorphism <img src="https://www.zhihu.com/equation?tex=%5Cvarphi%3A+G+%5Cto+%5Cgamma" alt="[公式]" data-formula="\varphi: G \to \gamma" /> is in <img src="https://www.zhihu.com/equation?tex=C%5Ek" alt="[公式]" data-formula="C^k" /> for <img src="https://www.zhihu.com/equation?tex=k+%5Cgeq1" alt="[公式]" data-formula="k \geq1" /> and the rank of <img src="https://www.zhihu.com/equation?tex=%5Cvarphi" alt="[公式]" data-formula="\varphi" /> is maximal (equal to 1), then we say this curve is <i>k-fold continuously differentiable</i>. For <img src="https://www.zhihu.com/equation?tex=k%3D1" alt="[公式]" data-formula="k=1" /> , we say that <img src="https://www.zhihu.com/equation?tex=%5Cgamma" alt="[公式]" data-formula="\gamma" /> is <i>smooth</i>.</p> <p><b>Definition 2</b> Let a smooth curve <img src="https://www.zhihu.com/equation?tex=%5Cgamma" alt="[公式]" data-formula="\gamma" /> be given by the parametric equations</p> <p><img src="https://www.zhihu.com/equation?tex=%5Cmathbf%7B%5Coverrightarrow%7Br%7D%7D+%3D+%5Cmathbf%7B%5Coverrightarrow%7Br%7D%7D%28t%29+%3D+x%28t%29%5Cmathbf%7B%5Coverrightarrow%7Bi%7D%7D%2By%28t%29%5Cmathbf%7B%5Coverrightarrow%7Bj%7D%7D%2Bz%28t%29%5Cmathbf%7B%5Coverrightarrow%7Bk%7D%7D.%5C%5C" alt="[公式]" data-formula="\mathbf{\overrightarrow{r}} = \mathbf{\overrightarrow{r}}(t) = x(t)\mathbf{\overrightarrow{i}}+y(t)\mathbf{\overrightarrow{j}}+z(t)\mathbf{\overrightarrow{k}}.\\" /> The <i>velocity vector</i> of <img src="https://www.zhihu.com/equation?tex=%5Cmathbf%7B%5Coverrightarrow%7Br%7D%7D%28t%29" alt="[公式]" data-formula="\mathbf{\overrightarrow{r}}(t)" /> at <img src="https://www.zhihu.com/equation?tex=t+%3D+t_0" alt="[公式]" data-formula="t = t_0" /> is the derivative</p> <p><img src="https://www.zhihu.com/equation?tex=%5Cmathbf%7B%5Coverrightarrow%7Br%7D%7D%27%28t_0%29+%3D+x%27%28t_0%29%5Cmathbf%7B%5Coverrightarrow%7Bi%7D%7D%2By%27%28t_0%29%5Cmathbf%7B%5Coverrightarrow%7Bj%7D%7D%2Bz%27%28t_0%29%5Cmathbf%7B%5Coverrightarrow%7Bk%7D%7D.%5C%5C" alt="[公式]" data-formula="\mathbf{\overrightarrow{r}}'(t_0) = x'(t_0)\mathbf{\overrightarrow{i}}+y'(t_0)\mathbf{\overrightarrow{j}}+z'(t_0)\mathbf{\overrightarrow{k}}.\\" />The <i>velocity vector field</i> is the vector function <img src="https://www.zhihu.com/equation?tex=%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t%29" alt="[公式]" data-formula="\overrightarrow{\mathbf{r}}'(t)" /> . The <i>speed</i> of <img src="https://www.zhihu.com/equation?tex=%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t%29" alt="[公式]" data-formula="\overrightarrow{\mathbf{r}}(t)" /> at <img src="https://www.zhihu.com/equation?tex=t+%3D+t_0" alt="[公式]" data-formula="t = t_0" /> is the length <img src="https://www.zhihu.com/equation?tex=%7C%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%7C" alt="[公式]" data-formula="|\overrightarrow{\mathbf{r}}'(t_0)|" /> of the velocity vector.</p> <p><b>Definition 3</b> The <i>tangent line</i> to a smooth curve <img src="https://www.zhihu.com/equation?tex=%5Cgamma" alt="[公式]" data-formula="\gamma" /> at the point <img src="https://www.zhihu.com/equation?tex=P+%3D+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_0%29" alt="[公式]" data-formula="P = \overrightarrow{\mathbf{r}}(t_0)" /> is the straight line through the point <img src="https://www.zhihu.com/equation?tex=P+%3D+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_0%29+%5Cin+%5Cgamma" alt="[公式]" data-formula="P = \overrightarrow{\mathbf{r}}(t_0) \in \gamma" /> in the direction of the velocity vector <img src="https://www.zhihu.com/equation?tex=%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29" alt="[公式]" data-formula="\overrightarrow{\mathbf{r}}'(t_0)" /> .</p> <h2>Tangent line and osculating plane of a curve</h2> <p>We let <img src="https://www.zhihu.com/equation?tex=d" alt="[公式]" data-formula="d" /> denote the length of a chord of a curve joining the points <img src="https://www.zhihu.com/equation?tex=P+%3D+%5Cgamma%28t_0%29" alt="[公式]" data-formula="P = \gamma(t_0)" /> and <img src="https://www.zhihu.com/equation?tex=P_1+%3D+%5Cgamma%28t_1%29" alt="[公式]" data-formula="P_1 = \gamma(t_1)" /> and <img src="https://www.zhihu.com/equation?tex=h" alt="[公式]" data-formula="h" /> denote the length of a perpendicular dropped from <img src="https://www.zhihu.com/equation?tex=P_1" alt="[公式]" data-formula="P_1" /> onto the tangent line to <img src="https://www.zhihu.com/equation?tex=%5Cgamma" alt="[公式]" data-formula="\gamma" /> at the point <img src="https://www.zhihu.com/equation?tex=P" alt="[公式]" data-formula="P" /> .</p> <p><b>Lemma 1</b> Let <img src="https://www.zhihu.com/equation?tex=%28x%28t%29%2Cy%28t%29%2Cz%28t%29%29" alt="[公式]" data-formula="(x(t),y(t),z(t))" /> be continuous in <img src="https://www.zhihu.com/equation?tex=t" alt="[公式]" data-formula="t" /> . Then,</p> <p><img src="https://www.zhihu.com/equation?tex=%5Clim_%7Bt_1+%5Cto+t_0%7D+%5B%28x%28t_0%29%2Cy%28t_0%29%2Cz%28t_0%29%29+%5Ctimes+%28x%28t_1%29%2Cy%28t_1%29%2Cz%28t_1%29%29%5D+%3D+%5Coverrightarrow%7B%5Cmathbf%7B0%7D%7D.%5C%5C" alt="[公式]" data-formula="\lim_{t_1 \to t_0} [(x(t_0),y(t_0),z(t_0)) \times (x(t_1),y(t_1),z(t_1))] = \overrightarrow{\mathbf{0}}.\\" /></p> <p><i>Proof</i>: Trivial and left to the reader. <img src="https://www.zhihu.com/equation?tex=%5Csquare" alt="[公式]" data-formula="\square" /></p> <p><b>Theorem 1</b></p> <p><img src="https://www.zhihu.com/equation?tex=%5Clim_%7Bd+%5Cto+0%7D+%5Cfrac%7Bh%7D%7Bd%7D+%3D+%5Clim_%7Bt_1+%5Cto+t_0%7D+%5Cfrac%7Bh%7D%7Bd%7D+%3D+0.%5C%5C" alt="[公式]" data-formula="\lim_{d \to 0} \frac{h}{d} = \lim_{t_1 \to t_0} \frac{h}{d} = 0.\\" /></p> <p><i>Proof</i>: We have that <img src="https://www.zhihu.com/equation?tex=d+%3D+%7C%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_1%29-%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%7C" alt="[公式]" data-formula="d = |\overrightarrow{\mathbf{r}}'(t_1)-\overrightarrow{\mathbf{r}}'(t_0)|" /> and by Proposition 2 that</p> <p><img src="https://www.zhihu.com/equation?tex=h+%3D+%5Cleft%7C%5Cfrac%7B%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%7D%7B%7C%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%7C%7D%5Ctimes+%5B%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_1%29-%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%5D%5Cright%7C.%5C%5C" alt="[公式]" data-formula="h = \left|\frac{\overrightarrow{\mathbf{r}}'(t_0)}{|\overrightarrow{\mathbf{r}}'(t_0)|}\times [\overrightarrow{\mathbf{r}}'(t_1)-\overrightarrow{\mathbf{r}}'(t_0)]\right|.\\" /></p> <p>We have, using properties of limits and keeping Lemma 1 in mind in the process,</p> <p><img src="https://www.zhihu.com/equation?tex=%5Cbegin%7Beqnarray%7D+%5Clim_%7Bd+%5Cto+0%7D+%5Cfrac%7Bh%7D%7Bd%7D%26%3D%26%5Clim_%7Bt_1%5Cto+t_0%7D%5Cleft%7C%5Cfrac%7B%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%7D%7B%7C%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%7C%7D%5Ctimes+%5Cleft%5B%5Cfrac%7B%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_1%29-%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%7D%7B%7C%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_1%29-%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%7C%7D%5Cright%5D%5Cright%7C%5C%5C+%26%3D%26%5Clim_%7Bt_1+%5Cto+t_0%7D+%5Cfrac%7B%7C%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%5Ctimes+%5Cfrac%7B%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_1%29-%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%7D%7Bt_1+-+t_0%7D%7C%7D%7B+%7C%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%7C%7C%5Cfrac%7B%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_1%29-%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%7D%7Bt_1+-+t_0%7D%7C%7D%5C%5C+%26%3D%26+%5Cfrac%7B%7C%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%5Ctimes+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%7C%7D%7B%7C%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%7C%5E2%7D+%3D+0.+%5Cend%7Beqnarray%7D%5C%5C" alt="[公式]" data-formula="\begin{eqnarray} \lim_{d \to 0} \frac{h}{d}&amp;=&amp;\lim_{t_1\to t_0}\left|\frac{\overrightarrow{\mathbf{r}}'(t_0)}{|\overrightarrow{\mathbf{r}}'(t_0)|}\times \left[\frac{\overrightarrow{\mathbf{r}}'(t_1)-\overrightarrow{\mathbf{r}}'(t_0)}{|\overrightarrow{\mathbf{r}}'(t_1)-\overrightarrow{\mathbf{r}}'(t_0)|}\right]\right|\\ &amp;=&amp;\lim_{t_1 \to t_0} \frac{|\overrightarrow{\mathbf{r}}'(t_0)\times \frac{\overrightarrow{\mathbf{r}}'(t_1)-\overrightarrow{\mathbf{r}}'(t_0)}{t_1 - t_0}|}{ |\overrightarrow{\mathbf{r}}'(t_0)||\frac{\overrightarrow{\mathbf{r}}'(t_1)-\overrightarrow{\mathbf{r}}'(t_0)}{t_1 - t_0}|}\\ &amp;=&amp; \frac{|\overrightarrow{\mathbf{r}}'(t_0)\times \overrightarrow{\mathbf{r}}'(t_0)|}{|\overrightarrow{\mathbf{r}}'(t_0)|^2} = 0. \end{eqnarray}\\" /></p> <p><img src="https://www.zhihu.com/equation?tex=%5Csquare" alt="[公式]" data-formula="\square" /></p> <p><b>Definition 4</b> A plane <img src="https://www.zhihu.com/equation?tex=%5Calpha" alt="[公式]" data-formula="\alpha" /> is called an <i>osculating plane</i> to a curve <img src="https://www.zhihu.com/equation?tex=%5Cgamma" alt="[公式]" data-formula="\gamma" /> at a point <img src="https://www.zhihu.com/equation?tex=P+%3D+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_0%29" alt="[公式]" data-formula="P = \overrightarrow{\mathbf{r}}(t_0)" /> if</p> <p><img src="https://www.zhihu.com/equation?tex=%5Clim_%7Bd%5Cto+0%7D+%5Cfrac%7Bh%7D%7Bd%5E2%7D+%3D+%5Clim_%7Bt_1+%5Cto+t_0%7D+%5Cfrac%7Bh%7D%7Bd%5E2%7D+%3D+0.%5C%5C" alt="[公式]" data-formula="\lim_{d\to 0} \frac{h}{d^2} = \lim_{t_1 \to t_0} \frac{h}{d^2} = 0.\\" /></p> <p><b>Theorem 2</b> At each point <img src="https://www.zhihu.com/equation?tex=P%3D+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_0%29" alt="[公式]" data-formula="P= \overrightarrow{\mathbf{r}}(t_0)" /> of a regular curve <img src="https://www.zhihu.com/equation?tex=%5Cgamma" alt="[公式]" data-formula="\gamma" /> of class <img src="https://www.zhihu.com/equation?tex=C%5Ek" alt="[公式]" data-formula="C^k" /> where <img src="https://www.zhihu.com/equation?tex=k+%5Cgeq+2" alt="[公式]" data-formula="k \geq 2" /> , there is an osculating plane <img src="https://www.zhihu.com/equation?tex=%5Calpha" alt="[公式]" data-formula="\alpha" /> , and the vectors <img src="https://www.zhihu.com/equation?tex=%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%2C%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%27%28t_0%29" alt="[公式]" data-formula="\overrightarrow{\mathbf{r}}'(t_0),\overrightarrow{\mathbf{r}}''(t_0)" /> are orthogonal to its unit normal vector <img src="https://www.zhihu.com/equation?tex=%5Coverrightarrow%7B%5Cboldsymbol%7B%5Cbeta%7D%7D" alt="[公式]" data-formula="\overrightarrow{\boldsymbol{\beta}}" /> .</p> <p><i>Proof</i>: Based on the following diagram from [1],</p> <figure data-size="normal"><noscript><img src="https://i2.wp.com/pic4.zhimg.com/v2-d4b39ec764b969ae82b0c141ec3e117f_b.jpg?w=327&#038;ssl=1" data-caption="" data-size="normal" data-rawwidth="327" data-rawheight="234" class="content_image" data-recalc-dims="1" /></noscript><img class="content_image lazy" src="data:;base64,<svg xmlns='http://www.w3.org/2000/svg' width='327' height='234'></svg>&#8221; width=&#8221;327&#8243; data-caption=&#8221;&#8221; data-size=&#8221;normal&#8221; data-rawwidth=&#8221;327&#8243; data-rawheight=&#8221;234&#8243; data-actualsrc=&#8221;<a href="https://pic4.zhimg.com/v2-d4b39ec764b969ae82b0c141ec3e117f_b.jpg&#038;#8221" rel="nofollow">https://pic4.zhimg.com/v2-d4b39ec764b969ae82b0c141ec3e117f_b.jpg&#038;#8221</a>; /></figure> <p>we have</p> <p><img src="https://www.zhihu.com/equation?tex=d+%3D+%7C%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_1%29+-+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_0%29%7C%2C+%5Cqquad+h+%3D+%7C%5Clangle+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_1%29+-+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_0%29%2C+%5Coverrightarrow%7B%5Cboldsymbol%7B%5Cbeta%7D%7D%5Crangle%7C.+%5Cqquad+%281%29%5C%5C" alt="[公式]" data-formula="d = |\overrightarrow{\mathbf{r}}(t_1) - \overrightarrow{\mathbf{r}}(t_0)|, \qquad h = |\langle \overrightarrow{\mathbf{r}}(t_1) - \overrightarrow{\mathbf{r}}(t_0), \overrightarrow{\boldsymbol{\beta}}\rangle|. \qquad (1)\\" /> We first prove the existence of osculating plane, for which there are two cases:</p> <ol> <li><img src="https://www.zhihu.com/equation?tex=%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29+%5Ctimes+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%27%28t_0%29+%5Cneq+0" alt="[公式]" data-formula="\overrightarrow{\mathbf{r}}'(t_0) \times \overrightarrow{\mathbf{r}}''(t_0) \neq 0" /> .</li> <li><img src="https://www.zhihu.com/equation?tex=%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29+%5Ctimes+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%27%28t_0%29+%3D+0" alt="[公式]" data-formula="\overrightarrow{\mathbf{r}}'(t_0) \times \overrightarrow{\mathbf{r}}''(t_0) = 0" /> .</li> </ol> <p>In the first case, we simply define the unit vector <img src="https://www.zhihu.com/equation?tex=%5Coverrightarrow%7B%5Cboldsymbol%7B%5Cbeta%7D%7D+%3D+%5Cfrac%7B%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_0%29+%5Ctimes+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_0%29%7D%7B%7C%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_0%29+%5Ctimes+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_0%29%7C%7D" alt="[公式]" data-formula="\overrightarrow{\boldsymbol{\beta}} = \frac{\overrightarrow{\mathbf{r}}(t_0) \times \overrightarrow{\mathbf{r}}(t_0)}{|\overrightarrow{\mathbf{r}}(t_0) \times \overrightarrow{\mathbf{r}}(t_0)|}" /> and in the second case, take any <img src="https://www.zhihu.com/equation?tex=%5Coverrightarrow%7B%5Cboldsymbol%7B%5Cbeta%7D%7D" alt="[公式]" data-formula="\overrightarrow{\boldsymbol{\beta}}" /> orthogonal to <img src="https://www.zhihu.com/equation?tex=%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29" alt="[公式]" data-formula="\overrightarrow{\mathbf{r}}'(t_0)" /> , which is non-zero by definition of regular curve. In both cases, we have</p> <p><img src="https://www.zhihu.com/equation?tex=%5Clangle+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%2C+%5Coverrightarrow%7B%5Cboldsymbol%7B%5Cbeta%7D%7D%5Crangle+%3D+%5Clangle+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%27%28t_0%29%2C+%5Coverrightarrow%7B%5Cboldsymbol%7B%5Cbeta%7D%7D%5Crangle+%3D+0.+%5Cqquad+%282%29%5C%5C" alt="[公式]" data-formula="\langle \overrightarrow{\mathbf{r}}'(t_0), \overrightarrow{\boldsymbol{\beta}}\rangle = \langle \overrightarrow{\mathbf{r}}''(t_0), \overrightarrow{\boldsymbol{\beta}}\rangle = 0. \qquad (2)\\" /> Let <img src="https://www.zhihu.com/equation?tex=%5Calpha" alt="[公式]" data-formula="\alpha" /> be the plane passing through the point <img src="https://www.zhihu.com/equation?tex=P+%3D+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_0%29" alt="[公式]" data-formula="P = \overrightarrow{\mathbf{r}}(t_0)" /> and orthogonal to <img src="https://www.zhihu.com/equation?tex=%5Coverrightarrow%7B%5Cboldsymbol%7B%5Cbeta%7D%7D" alt="[公式]" data-formula="\overrightarrow{\boldsymbol{\beta}}" /> . By Taylor&#8217;s formula,</p> <p><img src="https://www.zhihu.com/equation?tex=%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_1%29+-+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_0%29+%3D+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%28t_1+-+t_0%29+%2B+%5Cfrac%7B1%7D%7B2%7D%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%27%28t_0%29%28t_1+-+t_0%29%5E2+%2B+%5Coverrightarrow%7B%5Cboldsymbol%7Bo%7D%7D%28%7Ct_1+-+t_0%7C%5E2%29.+%5Cqquad+%283%29%5C%5C" alt="[公式]" data-formula="\overrightarrow{\mathbf{r}}(t_1) - \overrightarrow{\mathbf{r}}(t_0) = \overrightarrow{\mathbf{r}}'(t_0)(t_1 - t_0) + \frac{1}{2}\overrightarrow{\mathbf{r}}''(t_0)(t_1 - t_0)^2 + \overrightarrow{\boldsymbol{o}}(|t_1 - t_0|^2). \qquad (3)\\" /> Applying <img src="https://www.zhihu.com/equation?tex=%282%29%2C%283%29" alt="[公式]" data-formula="(2),(3)" /> to <img src="https://www.zhihu.com/equation?tex=%281%29" alt="[公式]" data-formula="(1)" /> gives</p> <p><img src="https://www.zhihu.com/equation?tex=h+%3D+%7C%5Clangle+%5Coverrightarrow%7B%5Cboldsymbol%7Bo%7D%7D%28%7Ct_1+-+t_0%7C%5E2%29%2C+%5Coverrightarrow%7B%5Cboldsymbol%7B%5Cbeta%7D%7D%5Crangle%7C%2C+%5Cqquad+d+%3D%7C%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%28t_1-t_0%29%2B%5Coverrightarrow%7B%5Cboldsymbol%7Bo%7D%7D%28%7Ct_1-t_0%7C%29%7C.%5C%5C" alt="[公式]" data-formula="h = |\langle \overrightarrow{\boldsymbol{o}}(|t_1 - t_0|^2), \overrightarrow{\boldsymbol{\beta}}\rangle|, \qquad d =|\overrightarrow{\mathbf{r}}'(t_0)(t_1-t_0)+\overrightarrow{\boldsymbol{o}}(|t_1-t_0|)|.\\" /> From this, one can verify without difficulty that</p> <p><img src="https://www.zhihu.com/equation?tex=%5Clim_%7Bt_1+%5Cto+t_0%7D+%5Cfrac%7Bh%7D%7Bd%5E2%7D+%3D+0.%5C%5C" alt="[公式]" data-formula="\lim_{t_1 \to t_0} \frac{h}{d^2} = 0.\\" /> For the other part of this theorem, we simply uses <img src="https://www.zhihu.com/equation?tex=%281%29" alt="[公式]" data-formula="(1)" /> and <img src="https://www.zhihu.com/equation?tex=%283%29" alt="[公式]" data-formula="(3)" /> on the limit equal to zero to deduce the desired orthogonality relations. The details will be left to the reader as an exercise. <img src="https://www.zhihu.com/equation?tex=%5Csquare" alt="[公式]" data-formula="\square" /></p> <p><b>References</b></p> <ul> <li>[1] <a class=" wrap external" href="https://link.zhihu.com/?target=https%3A//gmachine1729.wpcomstaging.com/%25e5%2590%258d%25e5%258d%2595-c%25d0%25bf%25d0%25b8%25d1%2581%25d0%25ba%25d0%25b8-lists/%25e5%2590%258d%25e5%258d%2595-c%25d0%25bf%25d0%25b8%25d1%2581%25d0%25ba%25d0%25b8-lists-%25e6%2595%25b0%25e5%25ad%25a6%25e5%2592%258c%25e7%2589%25a9%25e7%2590%2586%25e4%25b9%25a6-math-and-physics-books/victor-andreevich-toponogov-vladimir-rovenski-differential-geometry-of-curves-and-surfaces_-a-concise-guide-birkha%25cc%2588user-boston-2005/" target="_blank" rel="noopener noreferrer">Victor Andreevich Toponogov, Vladimir Rovenski &#8211; Differential Geometry of Curves and Surfaces: A Concise Guide (2005)</a></li> </ul> </div>
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