last modifications

Author: govin08

_posts/2025-03-17-wheels.md

@@ -179,7 +179,7 @@ $$\alpha = \frac{\beta_-+\beta_+}2$$
 $$\alpha'\perp\beta_-\beta_+$$
 
 이다.
-일반성을 잃지 않고 모든 $s$에 대하여 $\left|\left|\alpha'(s)\right|\right|=1$이라고 하고 $T(s)$를 $T(s)=\alpha'(s)$라고 정의하면 이것은 $\alpha$에 대한 통상적인 unit tangent vector가 된다.
+일반성을 잃지 않고 모든 $s$에 대하여 $\left|\left|\alpha'(s)\right|\right|=1$이라고 하고 $T(s)$를 $T(s)=\alpha'(s)$라고 정의하면 이것은 $\alpha$에 대한 통상적인 principal unit tangent vector가 된다.
 
 $$T\cdot T = \left|\left|T\right|\right|^2=1$$
 
@@ -188,7 +188,7 @@ $$T\cdot T = \left|\left|T\right|\right|^2=1$$
 $$T\cdot T' + T'\cdot T=0,$$
 
 즉, $T\cdot T'=0$이다.
-$T(s)$를 양의방향(시계반대방향)으로 회전한 벡터를 $N(s)$라고 하면 이것은 통상적인 unit normal vector가 되며, 앞서 논리와 마찬가지로 하면 $N\cdot N'=0$이다.
+$T(s)$를 양의방향(시계반대방향)으로 회전한 벡터를 $N(s)$라고 하면 이것은 통상적인 principal unit normal vector가 되며, 앞서 논리와 마찬가지로 하면 $N\cdot N'=0$이다.
 또한, $T(s)$와 $N(s)$는 평면의 orthonormal basis를 이루고 따라서
 
 $$T'(s)=\kappa(s)N(s)$$

_posts_hided/test.md

@@ -0,0 +1,39 @@
+# first
+
+Consider a plain $\mathbb R^2$.
+For some unit vector (that we would call 'the principal unit tangent vector') $T(s)$, we can rotate $T(s)$ by $\frac\pi2$ to get $N(s)$ (that we would call 'principal normal tangent vector').
+It is elementary to show that $T(s)\cdot T'(s)$ so that there exists a real number $\kappa(s)$ such that $T'(s)=\kappa(s)N(s)$.
+We call $\kappa$, the (signed) curvature.
+It follows that
+
+$$\kappa(s)=\kappa(s)N(s)\cdot N(s)=T'(s)\cdot N(s).$$
+
+And I'll do some computation, starting from $T(s)=(\cos\theta,\sin\theta)$.
+Surely, $\theta$ is a function of $s$.
+Then
+$$
+T'(s)=\frac{d\theta}{ds}(-\sin\theta,\cos\theta)
+$$
+
+# second
+
+Consider a plain $\mathbb R^2$.
+For some unit vector $T(s)$ (that we would call 'the principal unit tangent vector'), we can calculate $N(s)$ by
+
+$$
+N(s) = \frac{T'(s)}{\lvert T'(s)\rvert}
+$$
+
+that we would call 'the principal normal tangent vector'.
+It is elementary to show that $T(s)\perp T'(s)$ so that there exists a real number $\kappa(s)$ such that $T'(s)=\kappa(s)N(s)$.
+We call $\kappa(s)$, the (signed) curvature.
+It follows that
+
+$$\kappa(s)=\kappa(s)N(s)\cdot N(s)=T'(s)\cdot N(s).$$
+
+And I'll do some computation, starting from $T(s)=(\cos\theta,\sin\theta)$.
+Surely, $\theta$ is a function of $s$.
+Then
+$$
+T'(s)=\frac{d\theta}{ds}(-\sin\theta,\cos\theta)
+$$