Proof of the Envelope theorem for parametric functions in the Cartesian Plane Given a family of par

Dwllane4

Dwllane4

Answered question

2022-06-10

Proof of the Envelope theorem for parametric functions in the Cartesian Plane
Given a family of parametric functions in the Cartesian Plane (f(t,k); g(t,k)), their envelope is found by solving the equation:
f t g k f k g t = 0
How do I prove the above statement?

Answer & Explanation

lorienoldf7

lorienoldf7

Beginner2022-06-11Added 19 answers

Step 1
In the first place, you should never use the letter k for a continuous variable.
The condition you give concerns the Jacobian J Φ of the map
Φ : ( t , c ) ( f ( t , c ) , g ( t , c ) )   .
Step 2
If J Φ ( t 0 , c 0 ) 0 the map Φ maps a window with center ( t 0 , c 0 ) diffeomorphically onto a neighborhood of ( f ( t 0 , c 0 ) , g ( t 0 , c 0 ) ) , and this implies that the family of arcs
γ c : t ( f ( t , c ) , g ( t , c ) ) ( t 0 h < t < t 0 + h )
with c 0 δ < c < c 0 + δ looks like a family of parallel segments. It follows that the point p 0 := ( f ( t 0 , c 0 ) , g ( t 0 , c 0 ) ) cannot belong to the envelope of the given family of curves. Therefore your condition is necessary for envelope points. Sufficiency is another matter.

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