# How To Plot A Curve With Given Three Points?

1. Aug 8, 2011

### optics.tech

Hi everyone,

Can someone please tell me is there any mathematical equation/formula on ploting a curve with given three coordinates/points such as below image?

Thank you very much for your help

Op

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2. Aug 8, 2011

### hotvette

There are infinitely many curves you can draw through three points. The simplest method it to use a 2nd order polynomial (i.e a parabola). Just write out 3 equations using the values of x & y:

y1 = a x21 + b x1+ c
y2 = a x22 + b x2+ c
y3 = a x23 + b x3+ c

This is 3 equations in 3 unknowns (a,b,c) that can be solved by substitution.

3. Aug 8, 2011

### uart

As hotvette has already said, there are infinitely many curves you can draw through three points.

Two common cases in which the curve is uniquely specified by three points are the parabola and the circle. That is, if you assume it's a circle then such a circle is unique, and if you assume it's a parabola then such a parabola is unique.

4. Aug 9, 2011

### JJacquelin

No, there are an infinity of parabolas that can be draw through three points, each one with a different axial direction. Of course, il you asume a given direction for the axis, the parabola is unique.

5. Aug 9, 2011

### optics.tech

What about if the amount of the points are nine pieces?

Will above equations can be continued from three to nine equations with similar pattern?

Does this curve is a kind of smooth and continue curve?

6. Aug 10, 2011

### uart

If you want to extend this to an arbitrary number of points then you're probably looking for something more like a spline (piecewise fit). See: http://en.wikipedia.org/wiki/Spline_(mathematics)

Last edited: Aug 10, 2011
7. Aug 10, 2011

### agentredlum

If all 3 point's are colinear then it can't be a circle or a parabola, unless the radius of the circle goes to infinity or the coefficient of x^2 goes to infinity or zero for the parabola. It can be many other things but the best representation is probably a straight line, unless you have some kind of 'exotic' application such as temperature changes over time.

8. Aug 10, 2011

### mikeph

For nine points I think you'll need a 9th order polynomial, so you'd get nine equations (and a headache). Something tells me the reasoning stems from the fundamental theorem of algebra, but I couldn't give you more detail than that.