Proving Symmetry with Parametric FunctionI'm trying to brush up on some math for my latest Computer Science class. We are given a shape (curve) in parametric form. How can we show that this is symmetric (or not) about the x-axis or y-axis?If we have an equation in explicit form like: y = m x + bThen to show that this line is symmetric about some axis, we can set either x = − x or y = − y and show if the equation remains the same. However, if we have an equation in parametric form, such as: x ( t ) = s i n ( t ) , y ( t ) = c o s ( t )Then how do we go about showing if this resulting shape is symmetric or not? Is there a way to do so without converting back to the explicit form?

Question

Proving Symmetry with Parametric FunctionI&#039;m trying to brush up on some math for my latest Computer Science class. We are given a shape (curve) in parametric form. How can we show that this is symmetric (or not) about the x-axis or y-axis?If we have an equation in explicit form like:   y  =  m  x  +  bThen to show that this line is symmetric about some axis, we can set either   x  =  −  x or   y  =  −  y and show if the equation remains the same. However, if we have an equation in parametric form, such as:  x  (  t  )  =  s  i  n  (  t  )  ,  y  (  t  )  =  c  o  s  (  t  )Then how do we go about showing if this resulting shape is symmetric or not? Is there a way to do so without converting back to the explicit form?

Schetterai · Accepted Answer

Step 1Maybe a bit of a stupid but immediate method. Assume your function is given in parametric form  x  =  φ  (  t  )  y  =  ψ  (  t  )where   t  ∈  (  a  ,  b  ). Assume you want to check whether your curve is symmetric with respect to   x  ↦  −  x and   y  ↦  y, i.e. reflection in the y axis. For each value       t    0    ∈  (  a  ,  b  ) form the functionStep 2  f  (  t  ,      t    0    )  =            (        φ  (  t  )  +  φ  (      t    0    )                    )              2    +            (        ψ  (  t  )  −  ψ  (      t    0    )                    )              2  Observe   f  (  t  ,      t    0    )  ≥  0 and   f  (  t  ,      t    0    )  =  0 if and only if for some   t  ∈  (  a  ,  b  )  ,,       φ  (  t  )  =  −  φ  (      t    0    ) and   ψ  (  t  )  =  ψ  (      t    0    ). So if for each       t    0    ∈  (  a  ,  b  ) you can find a solution to the equation   f  (  t  ,      t    0    )  =  0 in terms of   t  ∈  (  a  ,  b  ) then your curve is symmetric with respect to the y axis.

Proving Symmetry with Parametric Function I'm trying to brush up on some math for my latest Compute

Answered question

Answer & Explanation

New Questions in Integral Calculus