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LucasGB
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The title says it all: can I pick a piece of paper, draw a completely random curve, and then describe it by an equation? Rephrasing, Can ANY curve be described by an equation?
Dragonfall said:You'll have to define "curve".
JSuarez said:No, what you have to define is what you mean by equation. In any case, the answer is no: the number of continuous curves is non-denumerable, while the number of (reasonable) ways to describe them is enumerable.
LucasGB said:I see, that's quite interesting! Can you link me to a proof of that statement?
Can you link me to a proof of that statement?
LucasGB said:JSuarez said:No, what you have to define is what you mean by equation. In any case, the answer is no: the number of continuous curves is non-denumerable, while the number of (reasonable) ways to describe them is enumerable.
I see, that's quite interesting! Can you link me to a proof of that statement?
CRGreathouse said:It's clear that there are uncountably many continuous curves, since there are uncountably many continuous curves of the form f(x) = k for k in R. It's clear that there are countably many formulas over a given alphabet, since they can be listed (order by length and then lexicographically).
The loophole in that argument is that, if we allow formulas with real valued coefficients, their number becomes uncountable
Yes:DaveC426913 said:Is there room in here for any curve being approximated to an arbitrary accuracy by an equation?
mathman said:Yes:
I'll assume the domain is [0,1] and the curve is a single valued function. For the nth approximation, evaluate the function at x=k/n for k between 0 and n. A polynomial can be fitted through these points. The approximation gets better as n increases.
LucasGB said:Wow, guys, this has got way too advanced for my understanding. What should I take from all of this? That if establish the condition that the equation must be finite, then there are more curves then equations, and not all curves can be described?
LucasGB said:The title says it all: can I pick a piece of paper, draw a completely random curve, and then describe it by an equation? Rephrasing, Can ANY curve be described by an equation?
some_dude said:I say no. For a random "curve" drawn by hand there exists absolutely no equation describing it (but arbitrarily close approximations). That would be akin to describing the relative position of every single atom you placed on the paper with your writing tool.
CRGreathouse said:Oh no, that's easy. There are only a finite number of atoms on the paper, so that curve can be described by a (parametric) polynomial. :P
some_dude said:Haha, okay I stand corrected. (If i knew a little more, i'd have some rebuttle referencing the Heisenberg uncertainty principle :P)
CRGreathouse said:Hey, if all you need to know is position, the Uncertainty Principle is no problem. :p
Yes; there is a theorem that says that the set of polynomials is a dense subset of the set of continuous functions on any closed interval. This is similar to the rationals (countable) being a dense subset of the reals (uncountable).DaveC426913 said:Is there room in here for any curve being approximated to an arbitrary accuracy by an equation?
Yes, any curve can be described by an equation. An equation is a mathematical representation that describes the relationship between different variables. As long as there is a relationship between the variables that can be expressed using mathematical operations, a curve can be described by an equation.
There are some limitations to describing curves with equations. Some curves, such as fractals, have complex and infinitely repeating patterns that cannot be fully described by a single equation. In these cases, multiple equations may be used to approximate the curve.
The type of equation used to describe a curve depends on the type of curve being described. Some common types of equations used for curves include linear equations, quadratic equations, exponential equations, and trigonometric equations.
No, not all curves can be described by a single equation. As mentioned before, some curves have complex and repeating patterns that cannot be fully captured by a single equation. In these cases, multiple equations or advanced mathematical techniques may be used to describe the curve.
In real-world applications, equations are used to describe curves in fields such as physics, engineering, and economics. They can be used to model and predict the behavior of systems, such as the trajectory of a projectile or the growth of a population. Equations are also used in computer graphics to create smooth and realistic curves in 3D modeling and animation.