Statistics How To

Implicitization: Simple Definition and Examples

Statistics Definitions > Implicitization

implicitizationIn mathematics, implicitization means making a variable which was explicit (for example, in a set of parametric equations) implicit.

If x = x (t) and y = y (t), implicitization would involve solving one of these equations for t, and then substituting that value of t into the other equation. You would end up with just one equation that had no mention of t. The idea is very similar to solving simultaneous equations in algebra, so if you can solve those, you can perform implicitization.

Examples of Implicitization

Suppose you wanted to implicitize x = a + b t and y = t2.

Step 1: Solve the first equation for t.

  • Subtract -a from both sides to get (x – a) = bt.
  • Divide by b, to get t= (x – a)/ b.

Step 2: Insert this into your second equation. y = t2, to end up with y = (x-a)2/b2.

This is the implicit equation; notice it doesn’t include the variable t.

For another example, let’s say you wanted to implicitize the set of equations y = a sin (t), x = a cos(t). Remember that cos(t)2 + sin(t)2 = (the Pythagorean trigonometric identity.) Since cos(t) = x/a and sin(t) = y/a, we rewrite the Pythagorean identity  as (x/a)2 + (y/a)2 = 1.

Not all implicitization problems are solvable. One example of a problem which can’t be implicitized into a single equation is the set of parametric equations x = α cos θ eβθ y = α sin θ eβθ. It’s a 2-parameter family of logarithmic spiral curves, but there is no one polynomial equation that describes it.

Higher Dimension Implicitization

Higher dimension implicitazation involves analyzing more than one parameter or more than two coordinates. In these cases you can use a Gröbner basis computation to implicitize. A Gröbner basis computation is essentially a generalization of Euclidean algebra as it relates to polynomials.

Sources

Ideals, Varieties, and Algorithms.
Geometric Computation.

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