From Math Images
Interesting page. A couple of comments.
First, is it possible to describe what kind of surface a parabolic reflector is? I'm guessing it's a paraboloid, but if you could include that somewhere in the more mathematical section (maybe before the proof), that would be great.
Also, three edits.
- should be replaced everywhere with
- In step 2, you write "The line normal to the parabola has the same slope relative to the y-axis as the line tangent to the parabola has relative to the x-axis, as shown in Figure 4." Replace "has the same slope" with "makes the same angle".
- Really, you ought to use different symbols for the theta in equation 1 and the theta in equation 2, and then show that these two symbols are equal, rather than using the symbol theta for both of them. You could use theta and phi, or theta_1 and theta_2 or something. Sorry, I know that's a pain to deal with.
Nice page with strong images and interesting applications.
Intro: Are you using parabolic and paraboloid synonymously? I think of parabolic as 2D and paraboloid as 3D.
Credits needed for diagrams.
- ⁋3: To me, an incoming beam changes its angle the least when it is high up the parabola because it is deflected only a little, whereas the beam near the vertex almost completely turns around. Obviously, it depends on which angle you are talking about, but it needs to be clearer.
Also, I would use "vertex" instead of "bottom" of the parabola. Notice how the EIA image does not have the vertex located at the bottom of the reflector.
- Step 2: "The slope of this tangent line is relative to the x-axis: when the slope is zero, the tangent line is parallel to the x-axis." Use a semi-colon instead of a colon to separate two complete ideas.
I get your idea, but I've never heard of having a slope relative to the y-axis. I might say that the angles are the same.
Step 5: I don't understand how you got from the fraction with x in the numerator to the fraction with 2tanθ in it. You say to use a trig identity but I can't figure it out.
Brendan, your paragraphs are getting smaller, good! However, they need to whenever appropriate have an associated image, and that image needs to be linked to the paragraph clearly. Help Keith and Maria develop the needed technology.
"a device to absorb the sun's energy" such as?
"the angle the light beam makes with the perpendicular when it hits the parabola is equal to the angle it makes with same perpendicular after being reflected." Please show.
Math Exp: requires elem calc and trig, right?
"The fact that a parabolic reflector can collect light in this way can be proven.", i.e. [mathematical statement of what you're going to prove] Then Step 1, etc.
This is a nice thing to have and your treatment is good, too. I hesitate to suggest this to you since it would be nice to give somebody else a chance with this important, fundamental stuff, but do you know the echo bench in the President's Garden? If two people sit in the right spots (with a fair degree of latitude) and whisper, they can hear each other pretty clearly--it's elliptical and sound bounces from one focus to the other.