贝塞尔曲线原理及其算法实现
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2022-06-17
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Bézier Curves 贝塞尔曲线
An Introduction 介绍
Our first question is: why do we need new forms of parametric curves? An immediate answer is
that those parametric curves discussed in the previous unit are not very geometric. More
precisely, given such a parametric form it is difficult to know the underlying geometry it
represents without some further analysis. The coefficients of the equations do not have any
geometric meaning, and it is almost impossible to predict the change of shape if one or more
coefficients are modified. As a result, designing a curve that follows certain outline is very
difficult.
我们的第一个问题是：为什么我们需要新形式的参数曲线？一个直接的答案是前面单元中讨
论的那些参数曲线不是很几何。更准确地说，给定这样的参数形式，如果不进行进一步分析，
就很难知道它所代表的基本几何形状。方程的系数没有任何几何意义，如果修改一个或多个
系数，几乎不可能预测形状的变化。因此，设计一条遵循特定轮廓的曲线非常困难。
In practice, designers or users usually do not care about the underlying mathematics (and
equations, of course). They are more or less focusing on getting their jobs done. To do so, a
system that supports users to design curves must be
在实践中，设计者或用户通常不关心底层数学（当然还有方程）。他们或多或少专注于完成
工作。为此，必须有一个支持用户设计曲线的系统
1. Intuitive 直观:
We expect that every step and every algorithm will have an intuitive and geometric
interpretation.
我们期望每一步和每一个算法都有一个直观和几何的解释。
2. Flexible 灵活:
The system should provide the users with more control for designing and editing the shape
of a curve. The way of creating and editing a curve should be easy, intuitive and geometric
rather than by manipulating equations.
系统应该为用户提供更多的控制来设计和编辑曲线的形状。创建和编辑曲线的方式应该
是简单、直观和几何的，而不是通过操纵方程。
3. Unified Approach 统一方法:
The way of representing, creating and editing different types of curves (e.g., lines, conic
sections and cubic curves) must be the same. That is, it does not require different
techniques for manipulating different curves (i.e., conics and cubics).
表示、创建和编辑不同类型曲线（如直线、圆锥曲线和三次曲线）的方式必须相同。也
就是说，它不需要不同的技术来处理不同的曲线（即圆锥曲线和三次曲线）。
4. Invariant 恒定:
The represented curve will not change its geometry under geometric transformations such
as translation, rotation and affine transformations.
表示的曲线在平移、旋转和仿射变换等几何变换下不会改变其几何形状。
5. Efficiency and Numerically Stability 效率和数值稳定性:
A user of a curve design system may not care about the beauty of the underlying geometry;
but, he/she expects the system to deliver the curve he/she wants fast and accurately.
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