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Lemniscate

Lemniscate. McKenzie Reimondo Emma Stoker Period B February 27, 2014. What is it?. A lemniscate graph takes on the shape of a figure eight It is similar to a rose graph, but it differs because a lemniscate graph only has two “petals”. Its Polar Equation. Equations of Lemniscates

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Lemniscate

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  1. Lemniscate McKenzie Reimondo Emma Stoker Period B February 27, 2014

  2. What is it? • A lemniscate graph takes on the shape of a figure eight • It is similar to a rose graph, but it differs because a lemniscate graph only has two “petals”.

  3. Its Polar Equation Equations of Lemniscates  where a is NOT equal to0

  4. How the a value affects the graph (cosine) • Positive a value vs. negative a value (in a cosine equation): The red graph has the equation: r= sqrt(2cos(2t)) The green graph has the equation: r= sqrt(-2cos(2t)) A positive a value makes the graph lie on the x-axis A negative a value flips the graph and makes it lie on the y-axis

  5. How the a value affects the graph (cosine) • a > 1 vs. a < 1 and larger vs. smaller a values that are > 1 (in a cosine equation): • The red graph equation: r=sqrt(.5cos(2t)) • The pink graph equation: r=sqrt(2cos(2t)) • The green graph equation: r=sqrt(3cos(2t)) • The blue graph equation: r=sqrt(4cos(2t)) • As the a value increases the loop of the petal expands and as the a value decreases the loop condenses

  6. How the a value affects the graph (sine) • Positive a value vs. negative a value (in a sine equation): • The black graph equation: r=sqrt(5sin(2t)) • The purple graph equation: r=sqrt(-5sin(2t)) • The positive a value makes the graph’s loops land in the first and third quadrant • The negative a value makes the loops land in the second and fourth quadrant

  7. How the a value affects the graph (sine) • a > 1 vs. a < 1 and larger vs. smaller a values that are > 1 (in a sin equation): • The green graph equation: r= sqrt(.5sin(2t)) • The red graph equation: r=sqrt(2sin(2t)) • The blue graph equation: r=sqrt(3sin(2t)) • The gray graph equation: r=sqrt(4sin(2t)) • The less the a value is the more condensed the graph is and the more the a value is the more the loops expand

  8. Cosine vs. Sine • Cosine lemniscate graphs lie on either the x or y-axis depending on the positive or negative a value. • Sine lemniscate graphs are on an angle and lie in two quadrants that are diagonal from each other; the positive or negative a value affects which two quadrants the graph lies in. • Both cosine and sine graphs go through the pole.

  9. Cites Used • http://sites.csn.edu/istewart/mathweb/math127/polar_equ/polar_equ.htm

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