What is a smooth surface in terms of tangents and normals? I read in a book that surfaces are smooth if its surface normals depend continuously on the points of that surface. I did not understand this definition, could somebody simplify it for me?
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$\begingroup$ the surface normal is a vector-function! $\endgroup$– W_DJul 21, 2013 at 19:27
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1$\begingroup$ so what is a smooth surface ? $\endgroup$– sagarJul 21, 2013 at 19:29
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$\begingroup$ The unit normal is a smooth vector function of the coordinate of the point on the surface so $n=(n_x,n_y,n_z)$ is a smooth function of $(x,y,z)$ that satisfies the equation of surface. A surface is generally parametrized through two variables, e.g., $r=(u,v)$, so you want $n$ to be a smooth function of $r$. The smooth typically means twice continuously differentiable. $\endgroup$– MaesumiJul 21, 2013 at 19:57
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$\begingroup$ Here is another way: hold a pencil perpendicular to an apple! Most of the time as you move the base of the point around the pen just moves smoothly with the base, but if you try to go through where the stems used to be the pencil goes through some jerky movements, at that point the surface of apple is not smooth! $\endgroup$– MaesumiJul 21, 2013 at 20:00
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1$\begingroup$ so when ever there is a sharp in the graph or a discontinuity where the normal does not exist , then the surface no longer remains smooth ? ! Did i get it right ? $\endgroup$– sagarJul 21, 2013 at 21:29
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4 Answers
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if you have a surface (in $\mathbb{R}^3$). it can be given by $x = x(u,v), y = y(u,v), z = z(u,v)$, where $u$ and $v$ are parameters from a subset $A\subset \mathbb{R}^2$. So $r(u,v) = (x(u,v),y(u,v),z(u,v))$ is a map from $A$ into $\mathbb{R}^3$. Whenever you fix any parameter, say, $u$, you'll obtain a curve on your surface, so, if it is possible (if these functions are differentiable) you can define tangent vertors $r_v$, $r_u$ (they are partial derivatives of $r$ with respect to respective parameters) for these curves. In some good cases these two vectors will not be parallel and thus will form the base of a tangent plane (but it doesn't always exist, consider the top of a cone) to your surface at any given point $(u,v)$. Then you can calculate the normal vector of unit length at any point $(u,v)$ by computing the cross product of $r_v$ and $r_u$ and dividing it by its length. In this way you'll obtain a function $n(u,v)$, it must be continuous... Sorry for being that unclear...
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$\begingroup$ The issue at the "top of a cone" is a discontinuity in the derivative. Also, you are considering a very narrow case, and some of the details are irrelevant. $\endgroup$ Jul 21, 2013 at 20:57
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"Smooth" just means that all the functions involved in the description of the object have infinitely many derivatives. In other words, they are continuous, differentiable, and so on for each derivative.
For instance, if your surface is described by polynomials, then it is smooth.
Note: To avoid confusion, when I say polynomial I mean polynomial, not "piecewise polynomial".
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$\begingroup$ You may wanna add that the polynomials shouldn't be a piecewisely defined one. $\endgroup$ Jul 21, 2013 at 21:13
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$\begingroup$ It seemed clear to me that "polynomial" doesn't mean "piecewise polynomial", but I guess it doesn't hurt to add this clarification. $\endgroup$ Jul 21, 2013 at 21:16
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A surface is a 2D subset of 3D. "IS" means there is a local homeomorphism (1 to 1 onto invertable relation) between 2D and the surface. Smoothnes is the continuity of this defining homeomorphism. If it is differentiable, the surface is SMOOTH or G1. G2 surfaces are aerodynamically desirable.
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Take an example of sphere. Now each and every point on sphere has a unique normal vector. That's why the surface of sphere is called smooth surface.
