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# 数学代写|曲线和曲面代写Curves And Surfaces代考|MATH2242 Further properties and examples

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## 数学代写|曲线和曲面代写Curves And Surfaces代考|Further properties and examples

A consequence of Property (iii) in the definition of $\beta$-spline functions is the affine invariance of the representation: if $A$ is an affine transformation, then the image $A(s)$ of a curve given by Formula (2.16) is represented by the control points $A\left(\boldsymbol{d}_i\right)$.

The convex hull property occurs if all $\beta$-spline functions are nonnegative, which depends on the connection parameters. In particular, if $\beta_{l, 1}=1$ and $\beta_{l, j}=0$ for $j>1$, then the $\beta$-spline functions are B-spline functions, which are nonnegative. Small enough perturbations of these particular connection parameters do not destroy this property, but it does not hold in general. If the functions $P_{k-n}^n, \ldots, P_k^n$ are nonnegative then the arc $\left{s(t): t \in\left[u_k, u_{k+1}\right]\right}$ is contained in the convex hull of the control points $\boldsymbol{d}_{k-n}, \ldots, \boldsymbol{d}_k$.

Figure $2.9$ shows two planar $\beta$-spline curves obtained with the knots and parameters that were used to define the cubic and quartic $\beta$-spline functions shown in Figure 2.8.

Figure $2.10$ shows two curves in the three-dimensional space: a cubic curve of class $G^2$ and a quartic curve of class $G^3$, with the same knots and connection parameters as the planar curves in the previous examples. The picture shows their curvature and torsion. To draw the picture, the Frenet frame has been found at a number of points of the curves and then the unit normal vector $\boldsymbol{n}$ was multiplied by the curvature $\kappa$ and a constant factor, and the binormal vector $\boldsymbol{b}$ was multiplied by the torsion $\tau$ and a constant factor. The curvature of both the curves is continuous. We can see points of discontinuity of the torsion of the cubic curve, while the torsion of the quartic curve is continuous.

The cubic curves in Figure $2.11$ have the same equidistant knots and the same (up to translations) control polygons; each of them was obtained with a different pair of global shape parameters, $\beta_1$ and $\beta_2$, which, for cubic curves, are named bias and tension. By looking at these curves one can see how the parameters influence the shape of the curve.

## 数学代写|曲线和曲面代写Curves And Surfaces代考|The turning tangents theorem

There is another very natural way of associating a $S^1$-valued curve (and consequently a degree) with a closed regular plane curve.

Definition 2.4.1. Let $\sigma:[a, b] \rightarrow \mathbb{R}^2$ be a closed regular plane curve of class $C^1$, and let $\mathbf{t}:[a, b] \rightarrow S^1$ be its tangent versor, given by
$$\mathbf{t}(t)=\frac{\sigma^{\prime}(t)}{\left|\sigma^{\prime}(t)\right|} .$$
The rotation index $\rho(\sigma)$ of $\sigma$ is the degree of the map $\mathbf{t}$; it counts the number of full turns made by the tangent versor to $\sigma$.

Corollary 2.1.18 provides us with a simple formula to compute the rotation index:

Proposition 2.4.2. Let $\sigma:[a, b] \rightarrow \mathbb{R}^2$ be a closed regular plane curve of class $C^1$ with oriented curvature $\tilde{\kappa}:[a, b] \rightarrow \mathbb{R}$. Then
$$\rho(\sigma)=\frac{1}{2 \pi} \int_a^b \tilde{\kappa}\left|\sigma^{\prime}\right| \mathrm{d} t=\frac{1}{2 \pi} \int_a^b \frac{\operatorname{det}\left(\sigma^{\prime}, \sigma^{\prime \prime}\right)}{\left|\sigma^{\prime}\right|^2} \mathrm{~d} t .$$
Proof. By Corollary 2.1.18,
$$\rho(\sigma)=\frac{1}{2 \pi} \int_a^b \operatorname{det}\left(\mathbf{t}, \mathbf{t}^{\prime}\right) \mathrm{d} t$$

## 数学代写|曲线和曲面代写Curves And Surfaces代考|The turning tangents theorem

$$\mathbf{t}(t)=\frac{\sigma^{\prime}(t)}{\left|\sigma^{\prime}(t)\right|} .$$

$$\rho(\sigma)=\frac{1}{2 \pi} \int_a^b \bar{\kappa}\left|\sigma^{\prime}\right| \mathrm{d} t=\frac{1}{2 \pi} \int_a^b \frac{\operatorname{det}\left(\sigma^{\prime}, \sigma^{\prime \prime}\right)}{\left|\sigma^{\prime}\right|^2} \mathrm{~d} t .$$

$$\rho(\sigma)=\frac{1}{2 \pi} \int_a^b \operatorname{det}\left(\mathbf{t}, \mathbf{t}^{\prime}\right) \mathrm{d} t$$

## MATLAB代写

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