Lecture 7: Intro to Geometry, Splines, and Bezier Curves (49)

rkumar06

If hermite splines are use for interpolation of to obtain a continuous function and the resulting spline will be continuous and will have continuous first derivative, why does it matter that hermite splines may have discontinuous second derivates? We discussed this be a differentiater between polynomials and hermite splines in discussion but I do not see how it relates as a pro or con.

gabeclasson

Visually, discontinuous second derivatives will look like sudden changes in inflection. I think it's kind of difficult to express, but this is something you can notice (see https://www.desmos.com/calculator/rh8aaix4q0 for example, which is continuous and has a continuous first derivative but to my eyes still feels unnatural at $x = 1$). I think it's honestly one of the distinctive qualities of all of the spline drawing tools you see in things like Microsoft Powerpoint, etc. It makes curves look slightly less "smooth," I guess is how I would describe it. But of course there are situations where you would want a discontinuous second derivative

Staffjamesfobrien

Curves are also used for things like a camera path where C2 discontinuities would be very noticeable. Also think about the specular reflections on the side of a nice, curvy sports car. The normals are based on 1st derivatives, so the specularities that are computed from the normals end up being functions of the derivatives. That means that c2 issues of the surface will show up as c1 issues in the specular reflections. A common way to try to spot evidence that a car was in an accident and not repaired perfectly is to look for jumps in the reflection as you look at where the body panels meet up.

Staffjamesfobrien

Curves are also used for things like a camera path where C2 discontinuities would be very noticeable. Also think about the specular reflections on the side of a nice, curvy sports car. The normals are based on 1st derivatives, so the specularities that are computed from the normals end up being functions of the derivatives. That means that c2 issues of the surface will show up as c1 issues in the specular reflections. A common way to try to spot evidence that a car was in an accident and not repaired perfectly is to look for jumps in the reflection as you look at where the body panels meet up.

If hermite splines are use for interpolation of to obtain a continuous function and the resulting spline will be continuous and will have continuous first derivative, why does it matter that hermite splines may have discontinuous second derivates? We discussed this be a differentiater between polynomials and hermite splines in discussion but I do not see how it relates as a pro or con.

Visually, discontinuous second derivatives will look like sudden changes in inflection. I think it's kind of difficult to express, but this is something you can notice (see https://www.desmos.com/calculator/rh8aaix4q0 for example, which is continuous and has a continuous first derivative but to my eyes still feels unnatural at $x = 1$). I think it's honestly one of the distinctive qualities of all of the spline drawing tools you see in things like Microsoft Powerpoint, etc. It makes curves look slightly less "smooth," I guess is how I would describe it. But of course there are situations where you would want a discontinuous second derivative

Curves are also used for things like a camera path where C2 discontinuities would be very noticeable. Also think about the specular reflections on the side of a nice, curvy sports car. The normals are based on 1st derivatives, so the specularities that are computed from the normals end up being functions of the derivatives. That means that c2 issues of the surface will show up as c1 issues in the specular reflections. A common way to try to spot evidence that a car was in an accident and not repaired perfectly is to look for jumps in the reflection as you look at where the body panels meet up.

Curves are also used for things like a camera path where C2 discontinuities would be very noticeable. Also think about the specular reflections on the side of a nice, curvy sports car. The normals are based on 1st derivatives, so the specularities that are computed from the normals end up being functions of the derivatives. That means that c2 issues of the surface will show up as c1 issues in the specular reflections. A common way to try to spot evidence that a car was in an accident and not repaired perfectly is to look for jumps in the reflection as you look at where the body panels meet up.