The Witch of Agnesi

Geometry Level 3

"The Witch of Agnesi" is a parametrically defined curve researched by Maria Agnesi in 1748.

What is the x ( t ) x(t) part of this parametric definition?

x ( t ) = ? ; y ( t ) = 2 sin ⁡ 2 ( t ) ; \hspace{20mm} x(t) = \ ?; \hspace{3mm} y(t) = 2\sin^2(t) \hspace{1mm};\hspace{3mm} for t = ( 0 , π ) t = (0, \pi)

Details:

x = 2 sin ⁡ ( t ) x=2\sin(t) x = 2 cot ⁡ ( t ) x=2\cot(t) x = cot ⁡ 2 ( t ) x=\cot^2(t) x = cos ⁡ ( t ) x=\cos(t)

Zandra Vinegar by Zandra Vinegar

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1 solution

Zandra Vinegar Staff
Oct 8, 2015

Since the circle has radius 1, the horizontal line above the circle must be y = 2 ( i n p u r p l e ) y=2 \hspace{2mm} \color{#69047E}{(in \hspace{2mm} purple)}

The line that begins at the bottom of the circle, at angle t t with the horizontal axis has slope = s i n ( t ) c o s ( t ) = t a n ( t ) = \frac{sin(t)}{cos(t)} = tan(t) and y-intercept 0. So the equation for the line is: y = t a n ( t ) x + 0 ( i n o r a n g e ) y = tan(t) \hspace{2mm} x + 0 \hspace{2mm} \color{#EC7300}{(in \hspace{2mm} orange)}

The x-coordinate of the intersection of these two lines ( o r a n g e a n d p u r p l e ) \color{#EC7300}{orange} \hspace{2mm} and \hspace{2mm} \color{#69047E}{purple}) defines the x-coordinate of the Witch of Agnesi curve ( g r e e n ) \color{#20A900}{green}) . So we have that 2 = t a n ( t ) x 2 = tan(t) \hspace{2mm} x

Solving for x x yields: x = 2 c o t ( t ) x = 2cot(t)

2 Helpful 0 Interesting 0 Brilliant 0 Confused

This curve is pretty awesome. It is now known to have applications in physics modeling the spectral line distribution of optical lines and x-rays, as well as the amount of power dissipated in resonant circuits.

Zandra Vinegar Staff - 5 years, 8 months ago

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