Editing Envelope (mathematics) (section)

===Example 2===
[[File:Envelope_string_art.svg|thumb|This plot gives the envelope of the family of lines connecting points (''t'',0), (0,''k'' - ''t''), in which ''k'' takes the value 1.]]
In [[string art]] it is common to cross-connect two lines of equally spaced pins. What curve is formed?

For simplicity, set the pins on the ''x''- and ''y''-axes; a non-[[orthogonal]] layout is a [[Coordinate rotation|rotation]] and [[scaling (geometry)|scaling]] away. A general straight-line thread connects the two points (0, ''k''&minus;''t'') and (''t'', 0), where ''k'' is an arbitrary scaling constant, and the family of lines is generated by varying the parameter ''t''. From simple geometry, the equation of this straight line is ''y'' = &minus;(''k''&nbsp;&minus;&nbsp;''t'')''x''/''t'' + ''k''&nbsp;&minus;&nbsp;''t''. Rearranging and casting in the form ''F''(''x'',''y'',''t'') = 0 gives:

{{NumBlk|:|<math>F(x,y,t)=-\frac{kx}{t} - t  + x + k -y = 0\,</math>|{{EquationRef|1}}}}

Now differentiate ''F''(''x'',''y'',''t'') with respect to ''t'' and set the result equal to zero, to get

{{NumBlk|:|<math>\frac{\partial F(x,y,t)}{\partial t}= \frac{kx}{t^2} - 1 = 0\,</math>|{{EquationRef|2}}}}

These two equations jointly define the equation of the envelope. From (2) we have:
: <math>t = \sqrt{kx} \,</math>
Substituting this value of ''t'' into (1) and simplifying gives an equation for the envelope:

{{NumBlk|:|<math>y=(\sqrt{x}-\sqrt{k})^2\,</math>|{{EquationRef|3}}}}

Or, rearranging into a more elegant form that shows the symmetry between x and y:

{{NumBlk|:|<math>\sqrt{x}+\sqrt{y}=\sqrt{k}</math>|{{EquationRef|4}}}}

We can take a rotation of the axes where the ''b'' axis is the line ''y=x'' oriented northeast and the ''a'' axis is the line ''y''=−''x'' oriented southeast. These new axes are related to the original ''x-y'' axes by {{math|1=''x''=(''b''+''a'')/{{sqrt|2}}}} and {{math|1=''y''=(''b''−''a'')/{{sqrt|2}}}} . We obtain, after substitution into (4) and expansion and simplification,

{{NumBlk|:|<math>b = \frac{1}{k\sqrt{2}} a^2 + \frac{k}{2\sqrt{2}},</math>|{{EquationRef|5}}}}

which is apparently the equation for a parabola with axis along ''a''=0, or ''y''=''x''.