How To Plot A Curve With Given Three Points?

In summary, the conversation discusses different methods for plotting a curve through three given points. Two common cases are a parabola and a circle, but there are infinitely many curves that can be drawn through three points. For an arbitrary number of points, a spline may be used. If all points are collinear, a straight line may be the best representation. For nine points, a 9th order polynomial may be used, but the reasoning behind it is not fully explained.
  • #1
optics.tech
79
1
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
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
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
if you assume it's a parabola then such a parabola is unique
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
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.

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
optics.tech said:
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?

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)
 
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  • #7
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.:smile:
 
  • #8
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.
 

1. How do I determine the equation of a curve with three given points?

To plot a curve with given three points, you can use the slope-intercept form of a line equation, y=mx+b, where m is the slope and b is the y-intercept. First, find the slope using the formula (y2-y1)/(x2-x1), where (x1,y1) and (x2,y2) are two of the given points. Then, substitute the slope and one of the points into the equation to find the value of b. Finally, use the equation to plot the curve on a graph.

2. Can I plot a curve with three points that are not collinear?

Yes, you can plot a curve with three points that are not collinear. However, keep in mind that the curve may not be a straight line, since three non-collinear points can form a unique curve.

3. Is it necessary to use the slope-intercept form of a line equation to plot a curve with three points?

No, it is not necessary to use the slope-intercept form of a line equation. Other forms of equations, such as point-slope form or standard form, can also be used to plot a curve with three points. However, the slope-intercept form is usually the most straightforward and commonly used method.

4. What if one of the given points is an outlier?

If one of the given points is an outlier, it may significantly affect the curve's shape and equation. In this case, it may be more accurate to plot the curve with the remaining two points and adjust the equation accordingly. Alternatively, you can use statistical methods, such as regression analysis, to determine the best-fit curve for the given points.

5. Are there any limitations to plotting a curve with three points?

Yes, there are limitations to plotting a curve with three points. The curve may not accurately represent the data if the points are not evenly spaced or do not follow a specific pattern. Additionally, using only three points may not provide enough information to determine the exact shape of the curve, especially if the points are relatively close to each other.

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