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Uberpuzzler

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 A Tangent to a Parabola   « on: Oct 29th, 2016, 8:01am » Quote Modify

A Tangent to a Parabola

In his beautiful work "Horologium Oscillatorium" (1673), Proposition 15, the Dutch mathematician and physicist Christiaan Huygens describes the mechanics of the straight edge and compass construction of a tangent to a cycloid. We are not as ambitious but, hopefully, as engaging:

given a graph of a parabola f(x) = ax2 and an arbitrary point P(x0, y0) on that graph, describe a procedure of a straight edge and compass construction of a tangent to the graph of f(x) passing through P if (given and orthogonal) axes Ox and Oy have no gradation marks on them (no line segment of unit length):

Extra for experts:

2) generalize

3) find the radius of a circle centered on Oy and tangent (internally) to the graph of f(x) at P

4) find the radius of the largest circle which, when rolled down one horn of the graph of f(x), will not get stuck near O but will rather roll right through it up and along the other horn

4') see if you can spot a connection between 4) and Off the Edge of the Earth
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towr
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 Re: A Tangent to a Parabola   « Reply #1 on: Oct 30th, 2016, 11:24am » Quote Modify

Maybe something like

draw y=x
construct line parallel to y=2x through intersection
construct focal-point of parabola (reverse of following steps)
draw line parallel to x=0 through P
draw line from focal point through P
bisect angle between previous two lines
draw line perpendicular to previous line through P
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Uberpuzzler

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 Re: A Tangent to a Parabola   « Reply #2 on: Oct 30th, 2016, 4:36pm » Quote Modify

I like it: a delicate step around the absence of a line segment of unit length plus, as a bonus, the location of parabola's focus.

towr's work is rendered here (the file name carries towr's credentials, the drawing is zoomed in a bit with P lowered with respect to the original, to avoid the clutter):

l1 = x, constructable, left as an exercise to the reader
l2 = 2x, constructable, ditto
l3
F - parabola's focus, constructable, left as an exercise to the reader (double the angle DAC)
l4 and l5, in the never ending chase for an optimization, the last step may be shaved off: the bisector of the angle FPB is it.

Nice non-analysis approach.

With one algorithm on the books, using an analysis-based argument, for an interpretation of a tangent, the number of steps may be reduced.
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towr
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 Re: A Tangent to a Parabola   « Reply #3 on: Oct 30th, 2016, 11:55pm » Quote Modify

Well, I wouldn't say it's completely non-analytic, after all I had to get it from somewhere that the tangent where y=x crosses y=ax^2 is parallel to y=2x

Reconsidering the morning after, I now realize you can cut out a significant number of steps because it's a matter of scaling one axis to make A=P
So
construct point Q = (x0, 2y0)
construct line through P parallel to OQ.
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Grimbal
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 Re: A Tangent to a Parabola   « Reply #4 on: Oct 31st, 2016, 3:43am » Quote Modify

Can we use the axes?  If so:

- build P' on the other horn such that OP = OP',
- A = the intersection of the y axis and PP',
- B = reflection of A thru O (i.e. on the negative y axis with OA = OB)
- PB is the tangent.

It is like towr's solution except that the line (0,0)-(x0,2y0) is replaced by (0,-y0)-(x0,y0).

 « Last Edit: Oct 31st, 2016, 3:50am by Grimbal » IP Logged
Uberpuzzler

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 Re: A Tangent to a Parabola   « Reply #5 on: Oct 31st, 2016, 8:28am » Quote Modify

@Grimbal: yes, absolutely. Grimbal's solution.

@towr: agreed. towr's second solution.

By analysis-based I meant the justification - and towr pretty much has it - instead of scaling along the Oy, scale along the Ox - from the right triangle ...
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SWF
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 Re: A Tangent to a Parabola   « Reply #6 on: Nov 6th, 2016, 1:46pm » Quote Modify

A generalization would be for any power, n, and the curve a*xn, draw the line through (0,-(n-1)*y0), as in Grimbal's solution. You don't even need the curve for the construction, just the point, origin, and one of the axes.
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Uberpuzzler

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 Re: A Tangent to a Parabola   « Reply #7 on: Nov 6th, 2016, 3:19pm » Quote Modify

Yes.

Correct.

By analytic justification I meant the textbookish observation that geometrically the first derivative is the slope (trigonometric tangent) of the tangent (straight line). Which means that in the right triangle PBX (first towr's drawing) where X is not shown - the point of intersection of the tangent and Ox BX = PB/tan(alpha) = ax2/2ax = x/2, etc.
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SWF
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 Re: A Tangent to a Parabola   « Reply #8 on: Nov 7th, 2016, 5:32pm » Quote Modify

One way to construct the focus is to make the line y=x/2.  Where it cross the parabola has the y coordinate of the focus, yF. This point is distance 2*yF from the focus and same distance from the directrix, while the origin is distance yF from both.
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Uberpuzzler

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 Re: A Tangent to a Parabola   « Reply #9 on: Nov 8th, 2016, 10:27am » Quote Modify

Nice.
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