Above is a picture of Joe's signature, where the units are in mm. Joe's signature is defined by the implicit equation
where we say the signature starts from the point
(
−
1
0
,
−
3
)
(on the script J) and ends at the point
(
2
6
,
−
9
)
(on the script e).
You can get the equation of Joe's signature in the text below.
Joe writes at a constant rate of 100 mm per sec. He notes that the time it takes for him to write his signature is less than 10 microseconds from a rational number of seconds, which can be written as
B
A
in lowest terms, where
A
and
A
+
B
are both perfect squares. Joe is happy because
A
+
A
+
B
is his age.
If Joe is younger than 100 years old, how old is he?
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This is not very easy to figure out though ... One way is to set x and y to polynomials of deg 15 in z with unknown coefficients and then plugging this into the implicit equation and solving for the unknowns by setting the coefficients of the resulting polynomial each to 0. You could impose extra conditions that say the curve starts at z=1 and ends at z=-1. (Of course you would need some software for this.)
Do you guys know of a more efficient way to figure out a parameterization (a priori one may not even exist ...)?
Do you guys know of a more efficient way to figure out a parameterization (a priori one may not even exist ...)?
Note that the Joe curve can be paramterized by
x(z)=-2 - 50 z + 18 z^2 - 368 z^3 - 1064 z^4 + 7120 z^5 + 7456 z^6 - 35648 z^7 - 22272 z^8 + 84224 z^9 + 34304 z^10 - 104448 z^11 - 26624 z^12 + 65536 z^13 + 8192 z^14 - 16384 z^15
y(z)=2 - 29 z - 48 z^2 + 560 z^3 - 1432 z^4 - 5328 z^5 + 11328 z^6 + 24256 z^7 - 32384 z^8 - 59392 z^9 + 45056 z^10 + 80896 z^11 - 30720 z^12 - 57344 z^13 + 8192 z^14 + 16384 z^15
for z from -1 to 1.
This is not very easy to figure out though ... One way is to set x and y to polynomials of deg 15 in z with unknown coefficients and then plugging this into the implicit equation and solving for the unknowns by setting the coefficients of the resulting polynomial each to 0. You could impose extra conditions that say the curve starts at z=1 and ends at z=-1. (Of course you would need some software for this.) Do you guys know of a more efficient way to figure out a parameterization (a priori one may not even exist ...)?