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October 25th, 2018, 11:38 AM   #11
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Quote:
Originally Posted by Arisktotle View Post
I guess the only escape from that is to assume that the meters do not correspond to the coordinates.
That was a confusing sentence. It should have read:

I guess the only escape from that is to assume that the coordinate values are not the same as the meters.

Last edited by Arisktotle; October 25th, 2018 at 11:44 AM.
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October 25th, 2018, 01:03 PM   #12
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The problem is complete as given.

y=ax^2 is the shape of the track. The parabola has to be shifted along x axis so that x=-100m (W) and y=100m (N) is a point on the track. Then ck the sign is outside the track and find the line from sign tangent to track (equate slope of line through sign to slope of a point on parabola.

EDIT:
To translate, y=a(x-k)^2 and -100,100 is a point on the curve gives k.

Last edited by zylo; October 25th, 2018 at 01:38 PM.
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October 25th, 2018, 01:58 PM   #13
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Quote:
Originally Posted by zylo View Post
The problem is complete as given.
Agree the problem is complete but have issues with the wind directions specified. Do not appear to be consistent with common conventions.
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October 25th, 2018, 02:18 PM   #14
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Quote:
Originally Posted by zylo View Post
To translate, y=a(x-k)^2 and -100,100 is a point on the curve gives k.
That doesn't work, as the given distances are supposed to be from the apex (i.e. the vertex) of the parabola.
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October 26th, 2018, 06:08 AM   #15
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Quote:
Originally Posted by skipjack View Post
That doesn't work, as the given distances are supposed to be from the apex (i.e. the vertex) of the parabola.
With vertex at the origin, rotate parabola about origin/vertex till it touches -100,100 and solve for angle of rotation. Exercise in analytic geometry.
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October 26th, 2018, 08:50 AM   #16
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Quote:
Originally Posted by zylo View Post
With vertex at the origin, rotate parabola about origin/vertex till it touches -100,100 and solve for angle of rotation. Exercise in analytic geometry.
ccw rotation t
x'=xcost+ysint
y'=-xsint+ycost
y'=ax'^2
Find t for case y=-x and then let x=-100.
-x(sint+cost)=ax^2(cost-sint)^2
(sint+cost)/(cost-sint)^2=-ax=18
Couldn't solve it.
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October 26th, 2018, 10:10 AM   #17
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There is a way to find rotation angle.

Let R,$\displaystyle \phi$ = (sqrt2)100,135 be the location of -100.100 in polar coordinates.
In polar coordinates, x=rcos$\displaystyle \theta$, y=rsin$\displaystyle \theta$ and y=ax^2 becomes
sin$\displaystyle \theta$=(ar)cos^2$\displaystyle \theta$=ar(1-sin^2$\displaystyle \theta$)
substitute a=.18, R=(sqrt2)100 and solve for $\displaystyle \theta$.
Then the rotation angle is 135-$\displaystyle \theta$

Now you can find equation of rotated track in x,y coordinates, ck sign is outside track, and find tangent to parabola from line through sign.

EDIT
Better yet. Solve the problem in the rotated coordinate system. Then all you need is the coordinates of the sign in the rotated coordinate system and y'=ax'^2.

Last edited by zylo; October 26th, 2018 at 10:33 AM.
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October 26th, 2018, 12:34 PM   #18
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Solving for $\displaystyle \theta$ gives that a radius of (sqrt2)100 intersects y=ax^2 at 78 and 102deg. To -100,100 the parabola has to be rotated 135-102 =33deg ccw. In a coordinate system rotated $\displaystyle \alpha$ = 33deg ccw the coordinates of the sign become:
x'=xcos$\displaystyle \alpha$+ysin$\displaystyle \alpha$=5cos33+4sin33=6.37
y'=ycos$\displaystyle \alpha$-xsin$\displaystyle \alpha$=4cos33-5sin33=.631

In the new coordinate system, dropping primes, the problem becomes
y=ax^2 with sign at 6.37,.631
At x=6.37 distance to track is .18(6.37)^2= 7.30 and sign is outside track.

Final step is to find line through sign tangent to track, and give result in original coordinate system. OP can do that.
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October 30th, 2018, 04:22 PM   #19
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The equation is broken because the sign is inside the parabola therefore it's never going to touch.

Last edited by skipjack; October 30th, 2018 at 11:23 PM.
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November 2nd, 2018, 06:08 PM   #20
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Equation of curve in rotated coordinate system: y=ax$\displaystyle ^{2}$
Coordinates of signal in rotated coordinate system: (r,s)=(6.37,.631) See my previous post.
Slope of curve = dy/dx = 2ax
Equation of straight line through signal: y=mx+b
s=mr+b $\displaystyle \rightarrow$ b=s-mr
y=mx+s-mr

This line is also tangent to curve at x$\displaystyle _{0}$, y$\displaystyle _{0}$ where m=2ax$\displaystyle _{0}$ and y$\displaystyle _{0}$=ax$\displaystyle _{0}^{2}$

$\displaystyle ax_{0}^{2}= (2ax_{0})x_{0}+s-2ax_{0}r$

This gives two tangent points from signal as expected. (solve for x$\displaystyle _{0}$ then y$\displaystyle _{0}$=ax$\displaystyle _{0}^{2}$)
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