# Paramatize a non-linear curve between points

• Tom McCurdy
In summary, there are an infinite number of non-linear curves that can pass through two given points. For a parabolic curve between (1,0) and (a,b), one possible parametrization is x= t, y= (b/(a^2-a)(t^2- t)), where a is not 1 or 0. However, there are still an infinite number of other curves that can pass through these points with different parametrizations.
Tom McCurdy
I was wondering how you could paramatize a non-linear curve between two points (1,0) and (a,b)

I have been trying to do like a parabolic paramatization but I am getting nowhere

I can get the first point alright

x=x
y=1-x^2
1<x<a
but then you get y=1-x^2

Any suggestions

do you now this?
x*lnx=9
x=?

EDIT:
To get a parabola going through (1,0) and (a,b), just set:
$$y=Ax^{2}+Bx+C$$
and determine A,BC by ther equations:
$$0=A+B+C, b=Aa^{2}+Ba+C$$

You'll get many parabolas satisfying your demands.

Last edited:
but only i need is a answer of that equation, how do you get that ''x'' out of there?

Tom McCurdy said:
I was wondering how you could paramatize a non-linear curve between two points (1,0) and (a,b)

I have been trying to do like a parabolic paramatization but I am getting nowhere

I can get the first point alright

x=x
y=1-x^2
1<x<a
but then you get y=1-x^2

Any suggestions

There is only one straight line between any two points but there are an infinite number of non-linear curves between any two points. If you want a parabolic curve between (1, 0) and (a,b), try this: x= t, y= u+ vx+ wt^2. In order that that pass through (1, 0) we must have x= 1 (so t= a), y= =u+ v+ w= 0. In order that it pass through (a, b) we must have x= a, (so t= a), y= u+ va+ va^2= b. That gives two equations for 3 unknown parameters so there are still an infinite number of such parabolic curves through those 2points. Select 1 by take u, say, equal to 0. Then v+ w= 0 and av+ a^2w= b. Multiply the first equation by a to get av+ aw= 0 and subtract: (a^2- a)w= b so w= b/(a^2- a), v= -1 = -b/(a^2-a). x= t, y= (b/(a^2-a)(t^2- t) works(as long as a is not 1 or 0). If a= 1 or 0, then a parabola with vertical axis will not pass through both points but some other parabola will and, of course, there are still an infinite number of other curves, each with an infinite number of possible parametrizations, that pass through (1, 0) and (a,b).

## 1. What does it mean to parametrize a non-linear curve between points?

Parametrizing a non-linear curve between points means to find a mathematical representation of the curve using a parameter or variable that varies between the given points. This allows for the curve to be described in terms of the parameter, making it easier to analyze and manipulate.

## 2. Why is it important to parametrize a non-linear curve?

Parametrizing a non-linear curve allows for easier analysis and manipulation of the curve, as well as making it possible to calculate important values such as slope, curvature, and area under the curve. It also allows for the curve to be graphed and compared to other curves more easily.

## 3. How do you parametrize a non-linear curve between points?

The process of parametrizing a non-linear curve involves finding a function that relates the points on the curve to a parameter. This can be done by solving for the parameter in terms of the coordinates of the points, or by using a standard parametric equation such as the parametric form of a circle or ellipse.

## 4. Can any non-linear curve be parametrized?

Yes, any non-linear curve can be parametrized using a suitable parameter or set of parameters. However, the parametrization may vary depending on the curve and the desired properties to be analyzed.

## 5. What are the advantages of parametrizing a non-linear curve?

The advantages of parametrizing a non-linear curve include being able to easily calculate important values, graph and compare the curve to other curves, and manipulate the curve to fit specific criteria. It also allows for a more intuitive understanding of the curve and its behavior.

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