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Family of Curves & Envelop Theorem

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Envelope Condition

Pencils refer to a family of curves that can be generated by varying a single parameter tt within an equation in the form F(x,y,t)=0F(x,y,t)=0.
Varying parameter tt can lead to different shapes and configurations within the family.
The collection of points (x,y)(x,y) that satisfy:
{F(x,y,t)=0∂F(x,y,t)∂t=0\begin{cases} F(x,y,t)=0 \\ \frac{ ∂F(x,y,t)}{∂t}=0 \end{cases}
form the envelope for the family of curves, F(x,y,t)=0F(x,y,t)=0.

Envelope of Lines

F(x,y,t)=yt+(11−t)(x−t)F(x,y,t)=yt+(11-t)(x-t)
Describes a family of lines.
t
F(x, y, t) = 0
0
x = 0
1
y = –10x + 10
2
2y = –9x + 18
3
3y = –8x + 24
4
4y = –7x + 28
5
5y = –6x + 30
6
6y = –5x + 30
7
7y = –4x + 28
8
8y = –3x + 24
9
9y = –2x + 18
10
10y = –x + 10
11
y = 0
∂F(x,y,t)∂t=∂∂t(yt+(11−t)(x−t))\frac{∂F(x,y,t)}{∂t}=\frac{∂}{∂t}(yt+(11-t)(x-t))
Solving for ∂∂t(yt+(11−t)(x−t))=0\frac{∂}{∂t}(yt+(11-t)(x-t))=0 yields:
−(x−y)2+22(x+y)−121=0-(x-y)^2+22(x+y)-121=0
Illustrated below is this envelope (red) of the family of linear curves (black).