I want to make a model of an ultrasound field, that impinges on a test object. The shape of the sound field can be simplified as a cone and the test object is cylindric. I used the following parametric equations:
Cone (with opening angle $\delta$): $$ x=t*\cos(\phi_{cone});\;y=t*\sin(\phi_{cone});\;z=-\frac{t}{\tan(\delta)}; $$ $$ r_{cone,top} \le t \le r_{cone,bot}; \;\; 0 \le \phi_{cone} \le 2\pi $$
Cylinder (with radius $a$): $$ x=b;\;y=a*\cos(\phi_{cylinder});\;z=a*\sin(\phi_{cylinder}); $$ $$ -l_{cylinder}/2 \le b \le l_{cylinder}/2; \;\; 0 \le \phi_{cylinder} \le \pi $$
To calculate the reflection and transmission, I thought of first finding the curve, where the two surfaces cut each other. Then I could calculate the angle between the sound field and the normal of the test objects surface. But I am somehow not able to find a parametric equation for the cut edge/curve. Is it even possible to solve this analytically? I added a picture to show, what it should look like.
Hope you can help me! =)
Let the cone $C$ be given by $$(t,\phi)\mapsto {\bf c}(t,\phi):=(t\cos\phi,t\sin\phi, \mu t)\qquad(t\geq0, \ 0\leq\phi\leq 2\pi)\ .$$ For each fixed $\phi$ we get a generator $g_\phi$ of the cone with parametric representation $$g_\phi:\quad t\mapsto (t\cos\phi,t\sin\phi, \mu t)\qquad(t\geq0)\ .$$ We now have to intersect $g_\phi$ with the cylinder $Z:\ y^2+z^2=a^2$. This amounts to solving $$t^2(\sin^2\phi+\mu^2)=a^2$$ for $t$. It follows that $g_\phi$ intersects the cylinder $Z$ when $$t={a\over\sqrt{\sin^2\phi+\mu^2}}\ ,$$ that is to say in the point $${\bf r}(\phi)={a\over\sqrt{\sin^2\phi+\mu^2}}(\cos\phi,\sin\phi,\mu)\qquad(0\leq\phi\leq2\pi)\ .\tag{1}$$ The equation $(1)$ is a parametric representation of the curve $\gamma:=C\cap Z$.
In order to determine the angle under which $C$ and $S$ intersect at a point ${\bf r}(\phi)$ we have to compute the normals of $C$ and $S$ at this point. For $C$ this amounts to computing ${\bf c}_t\times {\bf c}_\phi$ (and normalizing), for $S$ to compute $\nabla f$ at ${\bf r}(\phi)$ for the function $f(x,y,z):=y^2+z^2-a^2$.