最后给出一些实例来说明松弛参数和切向量对细分曲线的影响。
Finally, we illustrate the effects of using the tension parameter and tangent vectors to generate subdivision curves.
然后，可以用微分关系,或受约束的偏微分来表示,曲线的切向量，就是这样了。
Then actually there are ways you can use basically differentials and constrained partials to figure out what the tangent vector to the curve is and so on.
导出了线元切向量的变形方程以及拉伸轴在极射赤面投影图上位置变化计算公式。
The purely kinematical equation for the tangent vector of a line element is derived.
我所做的就是把F投影到切向量方向上，得出F沿着曲线的值,然后再把这些加起来。
What I do at any point is project F to the tangent direction, I figure out how much F is going along my curve and then I sum these things together.
这是另一种形式，它意味着,要求出每点处向量在切向量方向的分量,我还是用之前那个向量场吧。
That is how we reformulated it. That means we take our curve and we figure out at each point how big the tangent component I guess I should probably take the same vector field as before.
理解这一切的关键在于向量算术。
当然也有另一种方法，就是用参数方程表示这两条直线，用两条直线的方向向量作外积，从而得到切平面的法向量。
Another way to do it, of course, would provide actually parametric equations of these lines, get vectors along them and then take the cross-product to get the normal vector to the plane.
最好的方法是，既然我们已经有了梯度向量，就可以直接说,我们知道了切平面的法向量。
OK, the best way to it, now that we have the gradient vector, is actually to directly say oh, we know the normal vector to this plane.
那意味着，梯度向量在这点上，垂直于切平面或者是等值面。
Well, that means the gradient is actually perpendicular to the tangent plane or to the surface at this point.
这就是切平面的，或者说这个曲面的法向量。
OK, and that's going to be the normal vector to the surface or to the tangent plane.
求切平面方程的一般方法是先求切平面的法向量(借助偏导)，再用点法式写出切平面的方程。
The common method to quadric surface equation is to get the Normal Component first, and then write out the equation through dot method.
本文将在切触有理插值中起重要作用的Salzer定理推广到了多元向量的情形。
In this paper, the important Salzer's theorem for rational interpolation is generalized to the multivariate vector valued case.
第四章主要讨论了二元向量有理插值的迭加算法及二元向量切触有理插值的表现公式。
The chapter 4, we mainly discuss the overlay algorithm of two-variable vector-valued rational interpolation and show formula of two-variable vector-valued contact interpolation.
第四章主要讨论了二元向量有理插值的迭加算法及二元向量切触有理插值的表现公式。
The chapter 4, we mainly discuss the overlay algorithm of two-variable vector-valued rational interpolation and show formula of two-variable vector-valued contact interpolation.
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