Oktay Mercimek, EMAT 6690
Step 3 : Parabola Reflection Properties:
Now we can turn back to our parabola construction, and add circle trick to parabola8.gsp, and continue to explore reflection properties. As we know, light rays, which come from a far light source (like sun), are almost parallel to each other. We will use this information to explore reflection property of parabola.
Drag point B along the directrix for several incoming ray angles. I have created the incoming ray being parallel to segment HI, so when you move point B, direction of incoming ray will be always parallel to segment HI. This application is created to see any pattern between the direction of incoming ray and the locus of reflecting ray.
Do you see any pattern?
Now try this sketch which is similar to ParabolaReflecting1.gsp, but I added the locus of the Reflecting Ray.
Since locus of the Reflecting ray is available, you don't have to drag Point B. Now, drag point H and watch the changes in the locus. For what angle locus looks interesting?
When angleHIJ is 90 degrees, All reflecting rays intersect each other at Point P, focus of the parabola. As you know that, HI is parallel to Incoming Ray. That also means Incoming rays are perpendicular to the directrix of the parabola. That means if parabola mirror looks directly to the light source, it collects all light rays at the focus of the parabola. When AngleHIJ is different than 90 degrees, Reflecting Light rays don't intersect at a point, which means these angles are useless for us. It is also important to see that when AngleHIJ is different than 90 degrees, Reflecting Light rays doesn't even pass through the focus point, which means only Incoming light rays that are perpendicular to directrix pass through the focus.
Now we can try to understand why incoming light rays that are perpendicular to the directrix pass through directrix.
This is the parabola8 construction from the Step 1 Line BE, is one of the lines that we used during construction of the parabola, turned out to be correct direction to collect light rays. During construction we stated that Point D is the midpoint of the Segment BP and Line CD is perpendicular to the Segment BP. That means Triangle BPC is an isosceles triangle and m(AngleBCD)=m(AnglePCD). We also know that m(AngleGCF)=m(AnglePCD) and m(AngleBCD)=m(AngleECF). Therefore m(AngleECF)=m(AnglePCD). It means, if the Ray CE is the incoming ray to tangent line FD then Ray CP is the reflecting ray.
Now we have enough information to demonstrate Newton Telescope on the GSP.
Click Here to proceed Step 4 : Newton Telescope.