1. Finding Invariant Points from Linear Fractional Transformation (LFT)

A Linear Fractional Transformation (LFT), also known as a Möbius transformation, is given by:

$w = \frac{az + b}{cz + d}, \quad ad - bc \neq 0$

Invariant points are the points $z$ that remain unchanged under the transformation, i.e., $w = z$.

Steps:

1. Set $w = z$: $z = \frac{az + b}{cz + d}$

2. Multiply through by $cz + d$ (assuming $cz + d \neq 0$): $z(cz + d) = az + b$

3. Expand and simplify: $cz^2 + dz = az + b$

4. Rearrange into standard quadratic form: $cz^2 + (d - a)z - b = 0$

5. Solve this quadratic equation using the quadratic formula: $z = \frac{-(d - a) \pm \sqrt{(d - a)^2 + 4bc}}{2c}$

The two roots of this quadratic equation are the invariant points of the LFT.

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2. Parametric Representation of the Equation of a Curve

The parametric representation of a curve involves expressing the coordinates $(x, y)$ (and sometimes $z$ in 3D) as functions of a parameter $t$.

General Form:

For a 2D curve: $x = f(t), \quad y = g(t), \quad t \in \mathbb{R}$

For a 3D curve: $x = f(t), \quad y = g(t), \quad z = h(t), \quad t \in \mathbb{R}$

Example 1: Parametrize a Circle

The equation of a circle with radius $r$ centered at $(h, k)$ is: $(x - h)^2 + (y - k)^2 = r^2$

Parametric form: $x = h + r\cos(t), \quad y = k + r\sin(t), \quad t \in [0, 2\pi]$

Example 2: Parametrize a Line

The equation of a line in slope-intercept form $y = mx + c$: $x = t, \quad y = mt + c, \quad t \in \mathbb{R}$

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Questions for Further Exploration:

1. How do invariant points change when coefficients $a, b, c, d$ in LFT vary?
2. What are the geometric interpretations of invariant points for Möbius transformations?
3. How can parametric equations represent non-standard curves like ellipses or parabolas?
4. What methods exist to convert implicit curve equations into parametric form?
5. How do parametric representations simplify finding tangents and normals to a curve?

Tip:

When solving invariant points or parameterizing curves, always ensure the domain of the parameter $t$ is clearly defined to avoid ambiguities in the representation.