Locus of a variable point

• Additions and extensions
  
• Yet another example
  
  • Singular parts of a locus
    
  • Parasitic part of a locus
    
    • Example
      
  • Many ways
    
• Vectors and the parametric equation of a locus
  
  • Recall
    
  • The cycloid
    
  • The astroid
    

Additions and extensions

Yet another example

 
    P(xo,yo) is center-point
<=>
    Fx' (xo,yo,zo) = 0 and Fy' (xo,yo,zo) = 0
<=>
    P(xo,yo) is intersection point of the lines

    Fx' (x,y,z) = 0 and  Fy' (x,y,z) = 0

<=>
    P(xo,yo) is intersection point of the lines

    m x + y + m = 0 and x + m y - m = 0

 
    x2 - y2 + x + y = 0

<=>
    (x + y)(x - y + 1) = 0

Singular parts of a locus

Then, all points of these coinciding associated lines are points of the locus.

We say that these points form the singular part of the locus.

In previous example, the associated lines coincide for m = -1. The line (x - y + 1) = 0 is the singular part of the locus.

Parasitic part of a locus

Example

 
    x2 + y2 = 1

    y = m (x - 2)

 
(1 + m2) x2 - 4 m2 x + 4 m2 - 1 = 0

 
(1 + m2) y2 + 4 m y + 3 m2 = 0

The x-value of M is (x1 + x2)/2 = 2 m2/(1 + m2)

The y-value of M is (y1 + y2)/2 = - 2 m /(1 + m2)

So, we can say that M is the intersection point of the
associated lines

    x = 2 m2/(1 + m2)         (1)

    y = - 2 m /(1 + m2)        (2)

 
    (1)  <=>  (x - 2) m2 + x = 0   (3)

    (2)  <=>  y m2 + 2 m + y = 0   (4)

 
      (x - 2) m2 + x = 0      (3)

      (x-2) m - y = 0          (5)

 
    x2 + y2 - 2x = 0

<=>
    (x - 1)2 + y2 = 1

But, when you draw several instances of the variable line through point A, there are many positions of that line that don't generate points B and C. From this we see that the locus is only that part of the circle that is inside or on the circle x²+y²=1. Therefore, we say that all the other points of the locus are parasitic points.

Many ways

To illustrate this, we restart example 7 and we'll find the locus on a totally different way. Here is the problem again.

In an orthonormal coordinate system, a circle has equation x2 + y2 = 1 and A(2,0) is a fixed point. A variable line through point A, intersects the circle in points B and C. Calculate the locus of midpoint M of the segment [BC].

The variable line is y = m (x - 2).
The polar line of point P(1,m,0) with respect to the circle is x + my = 0 . This line cuts the variable line in point M.
The locus arises from eliminating m from these equations.
We find : x² + y² - 2x = 0.

Vectors and the parametric equation of a locus

Recall

• Take any two vectors P and Q, and call t the angle between P and Q.

  P.Q = ||P||.||Q||. cos(t)

• If E1 and E2 form an orthonormal basis in an orthonormal coordinate system. Say P(x,y) and Q(x',y') then

   
    P = Q  <=> (x,y) = (x',y') <=> (P.E1 = Q.E1  en  P.E2 = Q.E2)
  

The cycloid

Here is an illustration of the concept.

Choose the X-axis on the line l.
Choose the origin of the orthonormal coordinate system in point P, on the moment P is on the line l.

For each position of the circle, we have (vectors in bold)

 
    P = Q + QA + AP
<=>
    P.E1 = Q.E1 + QA.E1 + AP.E1
    P.E2 = Q.E2 + QA.E2 + AP.E2
<=>
    x = r.t + 0 + r.cos(-pi/2 - t)
    y =  0 +  r + r.cos(-pi - t)
<=>
    x = r(t - sin(t))
    y = r(1 - cos(t))

The astroid

Here is an illustration of the concept.

The small circle start rolling in point (R,0). Then t=s=0. Since the circle is rolling without slipping, we have that the two blue arcs have the same length.

 
   So, R.t = -R.s/4 <=>  s = - 4 t

   P = A + AP
=>
   P.E1 = A.E1 + AP.E1
<=>
    x = (3R/4).1.cos(t) + (R/4).1.cos(t+s)
<=>
    x = (3R/4).cos(t) + (R/4).cos(3t)             (1)

and in the same way

   P.E2 = A.E2 + AP.E2
=>
   y =  (3R/4).1.cos(pi/2 -t) + (R/4).1.cos(t - pi/2 + s)
<=>
   y =  (3R/4).sin(t) + (R/4).sin(3t)             (2)

Since cos(3t) = 4 cos3(t) - 3cos(t)
      sin(3t) = 3 sin(t) - 4 sin3(t)

we find the parametric equations of the astroid from (1) and (2)

    x = R cos3(t)
    y = R sin3(t)

 
    cos(t) = (x/R)1/3
    sin(t) = (y/R)1/3

=>  cos2(t) + sin2(t)  = (x/R)2/3 +  (y/R)2/3

<=>  x2/3 + y2/3 = R2/3