Problem 6: Envelopes

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• Direction of a tangent line of a curve.
  
• Curve associated with a set curves
  
• Envelope curve or the discriminant D
  
• Contact conditions
  
• A set of lines and its envelope
  
• Convex sets and support lines

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Direction of a tangent line of a curve.

• If the curve is given by an equation , then the slope of the tangent line is . A vector with this direction has coordinates or a scalar multiple of that ordered pair.

  Since , the direction of the tangent line at P = (x,y) is given by or .

• If the curve is given by an equation F(x,y) = 0, then is implicitly given by :

  

  So, gives the direction of the tangent line.

  If we have that F_x(x,y) = F_y(x,y) = 0 at a particular point Q = (x,y), then we say that Q is a singular point of the curve. The other points are regular points. If there are no singular points then we say that 0 is a regular value of F(x,y).

• If the curve is given by parameter equations x = f(t) ; y = g(t), then we have

  

  The direction of the tangent line at P is
  

Below we picture two examples of families of tangent lines to a function y = f(x).

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Curve associated with a set curves

Think of a curve C_t depending on a parameter t. For each t-value, the curve C_t takes a possibly different shape or position. If t varies in an interval, we have a 1-parameter family of curves.

Let's suppose that for each C_t we can associate just one point Q_t.

So with our 1-parameter family of curves C_t there corresponds a curve C: it is the locus of the points Q_t.

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Envelope curve

Two curves are tangent to each other if and only if both curves share a common tangent line at a common point.

The envelope curve of a set of curves C_t is an associated curve C such that the associated point Q_t is on C_t, and C is tangent to C_t at the point Q_t.

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Contact conditions - the case of regular points

The relation F(x,y,t) = 0 defines a 1-parameter family of curves C_t = F_t^-1(0) with parameter t - we assume that 0 is a regular value: this is necessary in order to guarantee that
C_t = F_t^-1(0) really are curves.

Note: F_t(x,y) is considered here as a function of x and y only.

Suppose that x = f(t) and y = g(t) are unknown parametric equations for the envelope curve C of the family: then the associated point Q_t has coordinates .

Since Q_t is on C_t we have

The tangent line to the curve C at point Q_t has the direction .

The tangent line to the curve C_t has the direction , and at the point Q_t that direction is - this direction is well-defined at a regular value of t.

For any given value of t, these directions must match. Thus,

(1) and (2) are the contact conditions.

Since (1) holds for each t, we get a new identity if we calculate the derivative, with respect to t, using the chain rule:

By virtue of this result (2) becomes

Thus (1) and (4) are the new and improved contact conditions.

With all this we can state: Let F(x,y,t) = 0 define the 1-parameter family of curves
C_t = F_t^-1(0). Say x = f(t) and y = g(t) are the parametric equations of an envelope curve C. Then the contact conditions are

This means that the parametric equations x = f(t) and y = g(t) of an envelope curve form a solution of the system

If x = f(t) and y = g(t) are the parametric equations of an envelope curve for the family of curves defined by F(x,y,t) = 0, these parametric equations form the discriminant set D and are a solution of the system (S)

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A set of lines and its envelope

As an example we take

As t varies, with x and y fixed, we have a set of lines. The parametric equations for envelope are solutions of the system

If we solve this system for x and y, we find:

These are the parametric equations of an ellipse, because if

,

The ellipse is a curve C such that C is tangent to every member of the set of lines:

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Convex sets and support lines

The -coordinate system is the one we will use most often for the space of lines on the plane, and it will have as its basic 2-form or density. In the figure above the line of interest is the support line to the given convex curve at the point P. A point (x,y) in the plane is on this support line if and only if

Since is really a function of when we are looking at the support lines to a convex curve, the equation above reads

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In order to find the envelope of all the support lines, which is the same thing as describing the convex curve, we need only solve simultaneously the system of equations S

to get the solutions

where

We will use this information in the next problem to find the area of a convex planar region as well as the length of its perimeter.