Edexcel A Level Further Maths: Core Pure:复习笔记2.2.3 Invariant Points & Lines

What is an invariant point?

• When applying transformations to a shape or collection of points, there may be some points that stay in their original position; these are known as invariant points

How can I find invariant points?

• If the point given by position vector , is invariant under transformation  then we can say that
  • 
• This will create a system of simultaneous equations which can be solved to find the invariant point
  • 
    • 
    • 
• The origin (0,0) is always invariant under a linear transformation

What is a line of invariant points?

• If every point on a line is mapped to itself under a particular transformation, then it is a line of invariant points
  • For example, a line of reflection is a line of invariant points

How can I find a line of invariant points?

• Use the same strategy as for finding a single invariant point:
  • If the point given by position vector , is invariant under transformation then we can say that
    • 
  • This will create a system of simultaneous equations which can be solved to find the invariant point(s)
    • 
    • 
    • 
• If there is a line of invariant points, rather than solving to find a single solution (a point), the two equations will be able to simplify to the same equation
  • This means that there are infinitely many solutions, and therefore infinitely many invariant points
    • A line contains infinitely many points
  • Your solution will be the equation of the invariant line e.g. y=3x

What’s the difference between a line of invariant points and an invariant line?

• If every point on a line is mapped to itself under a particular transformation, then it is a line of invariant points
  • Every single point on the line must stay in the same place

• With an invariant line however, every point on the line must simply map to another point on the same line
  • We are only concerned with the overall line; not the individual points

How do I find an invariant line?

• We can use a similar strategy to finding invariant points, with two slight changes
  • Use  to write the original position vector as
  • Write the transformed position vector as  using the same idea
    • Notice that the values of m and c will be the same, but different x and y coordinates
    • This because it is a different point, on the same line

• For an invariant line under transformation  we can write
  • 
• This will create a system of simultaneous equations which can be solved to find the invariant line(s)
  • 
  • 
  • The first equation can be substituted into the second to give an equation in terms of the variable x and the constants m and c
  • This equation can then be solved to find the values of m and c by equating the coefficients of x, and then equating the constant terms
    • There may be multiple solutions for m and c if there are multiple invariant lines