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# 数学代写|曲线和曲面代写Curves And Surfaces代考|MA3205 GEOMETRIC CONTINUITY AT A COMMON BOUNDARY

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## 数学代写|曲线和曲面代写Curves And Surfaces代考|GEOMETRIC CONTINUITY AT A COMMON BOUNDARY

Consider two patches represented by smooth regular parametrisations, $p(s, t)$ and $p^(u, v)$. We assume that the patches have a common boundary curve being a constant parameter curve of the first patch, and corresponding to $t=t_0$, and a constant parameter curve of the second patch, corresponding to $v=v_0$. Let $I$ denote the line segment $v=v_0$, bounding the domain of the parametrisation $p^$. We assume that the two parametrisations of the common curve, obtained by restricting the parametrisations $\boldsymbol{p}$ and $\boldsymbol{p}^$ and denoted by $\overline{\boldsymbol{p}}$ and $\underline{p}^$, are identical: $\overline{\boldsymbol{p}}(s)=$ $\underline{p}^(u)$ for $s=u$. This assumption guarantees the positional continuity of the junction of the two patches: $$\bar{p}=\underline{p}^ .$$
The derivation of equations of geometric continuity for a junction of two patches is similar to that of curve arcs. Using a function $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$, whose coordinates are described by scalar functions $s$ and $t$ (see Figure 3.1), we obtain a new parametrisaton of the patch $p$ :
$$q(u, v)=p(s(u, v), t(u, v)) .$$
We assume that the partial derivatives of $\boldsymbol{q}$ up to the order $n$ at each point of the line segment $I$ are equal to the corresponding derivatives of the parametrisation $p^*$.

We can obtain the derivatives of the parametrisation $\boldsymbol{q}$ using the generalised Fàa di Bruno’s formula (A.55) for functions of two variables. Then, we restrict them to the line segment $I$. We can notice that if the derivatives with respect to $v$ of $\boldsymbol{q}$ and $p^$ of any order $k$ are equal at each point of the line segment $I$, then also $$\left.\frac{\partial^{m+k}}{\partial u^m \partial v^k} \boldsymbol{q}\right|{v=v_0}=\left.\frac{\partial^{m+k}}{\partial u^m \partial v^k} \boldsymbol{p}^\right|{v=v_0}$$

for all $m$ such that these derivatives exist. Therefore, we can focus our attention on the partial derivatives with respect to $v$ – the parameter changing across the boundary-which is why they are called the cross-boundary derivatives of the patches.

The cross-boundary derivatives of the parametrisation $q$ are related to those of $p$ in the following way:
\begin{aligned} \overline{\boldsymbol{q}}v &=\bar{s}_v \overline{\boldsymbol{p}}_s+\bar{t}_v \overline{\boldsymbol{p}}_t, \ \overline{\boldsymbol{q}}{v v} &=\bar{s}{v v} \overline{\boldsymbol{p}}_s+\bar{t}{v v} \overline{\boldsymbol{p}}t+\bar{s}_v^2 \overline{\boldsymbol{p}}{s s}+2 \bar{s}v \bar{t}_v \overline{\boldsymbol{p}}{s t}+\bar{t}v^2 \overline{\boldsymbol{p}}{t t} \end{aligned}
etc. The general (and rather impractical) formula, which is a special case of (A.55), is
$\frac{\partial^j}{\partial v^j} \overline{\boldsymbol{q}}=\sum_{k=1}^j \sum_{h=0}^k a_{j k h} \frac{\partial^k}{\partial s^h \partial t^{k-h}} \overline{\boldsymbol{p}}$,
$a_{j k h}=\left(\begin{array}{l}k \ h\end{array}\right) \sum_{\substack{m_1+\cdots+m_k=j \ m_1, \ldots, m_k>0}} \frac{j !}{k ! m_{1} ! \ldots m_{k} !} \bar{s}{v^{m_1}} \ldots \bar{s}{v^{m_h}} \bar{t}{v^{m{h+1}} \ldots} \ldots \bar{t}_{v^{m_k}}$.

## 数学代写|曲线和曲面代写Curves And Surfaces代考|INTERPRETATION

We take a closer look at the junctions of patches of class $G^n$ for $n=1$ and $n=2$, bearing in mind that the patches have a common curve $\overline{\boldsymbol{p}}=\underline{p}^$. From the assumption that $\overline{\boldsymbol{p}}(s)=\underline{p}^(u)$ if $s=u$ it follows that all partial derivatives of the parametrisations $\boldsymbol{p}$ and $\boldsymbol{p}^*$ with respect to $s$ and $u$ at the junction points agree.

Case $n=1$. At any point of the common curve the cross-boundary derivative of $p^*$ is a linear combination of the first-order partial derivatives of $\boldsymbol{p}$. The partial derivatives of both patches at any point of their common curve determine the same plane (see Figure 3.2). Geometrically $G^1$ continuity is the continuity of tangent plane of the surface made of the two patches. One can also talk about the continuity of the normal vector, which is equivalent.
The same geometric interpretation applies to the equations for the homogeneous representations of the patches. The homogeneous patches reside in the four-dimensional space. At any junction point the triples of vectors, $\overline{\boldsymbol{P}}(s), \overline{\boldsymbol{P}}_s(s), \overline{\boldsymbol{P}}_t(s)$ and $\underline{\boldsymbol{P}}^(u), \underline{\boldsymbol{P}}_u^(u), \underline{\boldsymbol{P}}_v^*(u)$ span the same three-dimensional linear subspace (i.e., hyperplane) $\Pi(u)$ of $\mathbb{R}^4$. The common tangent plane $\pi(u)$ of the rational patches is represented by this hyperplane. ${ }^1$

Case $n=2$. The first and second fundamental forms (see Section A.10.2) are expressed by the derivatives of the first and second order of the surface’s parametrisation. The forms may be used to find the curvature of curves obtained by intersecting the surface with planes. If there exists a regular parametrisation of class $C^2$ of the surface, e.g. described piecewise by $q$ and $p^*$, then the curvature of the intersection of the surface with any plane not tangent to the surface is continuous.

On the other hand, having a surface whose all planar sections (with non-tangent planes) are curves with the curvature continuous, it is possible to find local regular parametrisations of class $C^2$ of this surface. ${ }^2$ Thus geometric continuity of the second order is equivalent to the curvature continuity of the surface.

## 数学代写|曲线和曲面代写Curves And Surfaces代考|GEOMETRIC CONTINUITY AT A COMMON BOUNDARY

$$\bar{p}=\underline{p}$$

$$q(u, v)=p(s(u, v), t(u, v)) .$$

$a_{j k h}=(k h) \sum_{m_1+\cdots+m_k=j m_1, \ldots, m_k>0} \frac{j !}{k ! m_{1} ! \ldots m_k} \bar{s} v^{m_1} \ldots \bar{s} v^{m_h} \bar{t} v^{m h+1} \ldots \ldots \bar{t}_{v^m k}$.

## 数学代写曲线和曲面代写Curves And Surfaces代考|INTERPRETATION

${ }^2$ 因此二阶几何连续性等价于曲面的曲率连续性。

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