TU Wien:Geometrie für Informatik VU (Raffaelli)/2022-02-04 2. Prüfung

Exercise 1

Given real constants $r\neq 0$ and $p$, consider the curve $\gamma :[0,1]\rightarrow \mathbb {R} ^{3}$ defined by $\gamma (t)=(r\;\cos(t),r\;\sin(t),pt).$

1. Determine the length of $\gamma$.
2. Determine two unit vectors $n_{1}(t),n_{2}(t)$ that are orthogonal to the curve's tangent $\gamma '(t)$ and such that $\langle n_{1}(t),n_{2}(t)\rangle =0$.

Solution

1.1:
Length $s(t)=\int _{t_{0}}^{t_{1}}||\gamma '(t)||dt$
$\gamma '(t)=(r\cdot -\sin(t),r\cdot \cos(t),p)$, $||\gamma '(t)||={\sqrt {(r\cdot -\sin(t))^{2}+(r\cdot \cos(t))^{2}+p^{2}}}={\sqrt {r^{2}\cdot \sin ^{2}(t)+r^{2}\cdot \cos ^{2}(t)+p^{2}}}={\sqrt {r^{2}\cdot (\sin ^{2}(t)+\cos ^{2}(t))+p^{2}}}={\sqrt {r^{2}+p^{2}}}$, Note: $(\cos ^{2}(v)+\sin ^{2}(v))=1$.
$s(t)=\int _{0}^{1}{\sqrt {r^{2}+p^{2}}}dt=t\cdot {\sqrt {r^{2}+p^{2}}}{\Big |}_{0}^{1}={\sqrt {r^{2}+p^{2}}}$

Exercise 2

Show that the surface patch $\sigma (u,v)=(u*cos(v),u*sin(v),v)$, where $(u,v)$ in $R^{2}$ is regular. Prove that the angle between the u- and v-curves is $\pi /2$.

Solution

Regular: partial derivatives must be linearly independent
$\sigma _{u}(u,v)=(cos(v),sin(v),0)$,
$\sigma _{v}(u,v)=(u*-sin(v),u*cos(v),1)$
Cross product: $(sin(v),-cos(v),u*(cos^{2}(v)+sin^{2}(v))$, Note: $(cos^{2}(v)+sin^{2}(v))=1$

(Note: Two vectors in $R^{3}$ are linearly dependent iff their cross product equals $(0,0,0)$, therefore $\sigma _{u}$ and $\sigma _{v}$ are linearly independent, as the cross product is not equal to $(0,0,0)$). Use dot product for better visibility: $sin^{2}(v)+cos^{2}(v)+u^{2}=1+u^{2}$. No matter what value u takes, it will be a positive value when it is squared. 1 + a positive value is larger than 0 --> patch is regular.
Angle: Calculate the dot product between the partial derivatives and show that it is zero.

Exercise 3

Single choice-exercise:

• A geodesic has zero normal curvature (FALSE)
• The normal curvature does not change under isometries (FALSE)
• A curve with no torsion is necessarily planar (TRUE)
• By definition, a regular surface patch is injective (FALSE)
• Every Monge patch is regular (TRUE)
• A local isometry between two surfaces is an angle-preserving map (TRUE)
• If a surface has zero Gaussian curvature, then it is planar (FALSE)
• Any local isometry preserves both Gaussian and mean curvature (FALSE)
• Every smooth curve has a unit-speed reparametrization (FALSE)
• A cone is an example of a developable surface (TRUE)

Exercise 4

Given a surface patch $sigma:R^{2}-->R^{3}$, $\sigma (u,v)=((u+v)/2,(v-u)/2,uv)$, compute the normal curvature of the surface curve corresponding to the straight line u = v in the parameter plane.