Finding Loci of Centres in Ellipses: Can You Help?

  • Thread starter Thread starter gianeshwar
  • Start date Start date
  • Tags Tags
    Ellipse
AI Thread Summary
The discussion focuses on finding the loci of various centers of triangles formed by a point on an ellipse and its two foci. Participants clarify the correct terminology, noting that "locus" is singular and "loci" is plural. They provide equations for the ellipse and suggest averaging the coordinates of the triangle's vertices to find the centroid locus, which results in another ellipse. There is confusion regarding the incenter locus, where one participant initially calculated a parabola, which was deemed incorrect. Ultimately, the correct locus for the incenter is confirmed to be an ellipse, aligning with the geometric properties of the triangle formed.
gianeshwar
Messages
225
Reaction score
14
20150814085536.jpg
Dear Friends!
If we have a variable triangle formed by joining a point of ellipse and its two focii shown in figure,how to find locii of various centres.Can you please suggest some reference to find these .I have tried by using simple method for incentre locus which I left as it goes long. Though I am still thinking.
 
Mathematics news on Phys.org
gianeshwar said:
View attachment 87310 Dear Friends!
If we have a variable triangle formed by joining a point of ellipse and its two focii shown in figure,how to find locii of various centres.Can you please suggest some reference to find these .I have tried by using simple method for incentre locus which I left as it goes long. Though I am still thinking.
First off, locus is singular and loci (one 'i') is plural, not "locii."

For your ellipse, the two foci are at ##(\pm c, 0)##, where ##c = \sqrt{a^2 - b^2}##). Possibly you could write an equation for the locus of the various centers as the point P on the ellipse moves around.
 
The center of such a triangle is the point whose coordinates are the means of the corresponding coordinates of the vertices.

Here, you have a ellipse given by \frac{x^2}{a^2}+ \frac{y^2}{b^2}= 1. Assuming that a> b, the two foci have coordinates (-\sqrt{a^2- b^2}, 0) and (\sqrt{a^2- b^2}, 0). You can write a point on that ellipse as P= (a cos(t), b sin(t) for parameter t between 0 and 2\pi. "Average" those coordinates.

(That turns out to be remarkably simple and gives a very easy answer!)
 
Last edited by a moderator:
Thanks Mark44 and Hallsoflvy!
 
HallsofIvy said:
The center of such a triangle is the point whose coordinates are the means of the corresponding coordinates of the vertices.

Here, you have a ellipse given by \frac{x^2}{a^2}+ \frac{y^2}{b^2}= 1. Assuming that a> b, the two foci have coordinates (-\sqrt{a^2- b^2}, 0) and (\sqrt{a^2- b^2}, 0). You can write a point on that ellipse as P= (a cos(t), b sin(t) for parameter t between 0 and 2\pi. "Average" those coordinates.

(That turns out to be remarkably simple and gives a very easy answer!)
 

Attachments

  • 20150819054530.jpg
    20150819054530.jpg
    26.5 KB · Views: 439
Dear friends ! In the solution of finding locus of incentre ,I am getting parabola as locus which does not seem to be by physical picturization.
 
Well, what did you do when you followed our suggestions? What is the "average" of the three points (-\sqrt{a^2- b^2}, 0), (\sqrt{a^2- b^2}, 0), and (a cos(t), b sin(t))? It is NOT a parabola! Please show exactly what you did.
 
Thanks Hallsfloy! I believe you are telling to find centroid locus which I am showing now. It is an ellipse which is fine.The first locus which I had calculated was of incentre in attached picture earliar.
 

Attachments

  • 20150819103735.jpg
    20150819103735.jpg
    49 KB · Views: 469
Yes, that's right. As I said originally, "That turns out to be remarkably simple and gives a very easy answer!" The y coordinates of the first two points are 0 and their x coordinates cancel so the equation of the locus is just the original ellipse, divided by 3. It is the ellipse with major and minor axes 1/3 the length of the original ellipse.
 
  • #10
Thanks! Now my doubt in #6 is still left!
 
  • #11
What "doubt" do you mean? The difficulty you stated in #6 was that you got a parabola which, of course, can't be right because a parabola would have to go outside the original ellipse. But in #8 you say you got an ellipse as the answer.
 
  • #12
Incenter is the intersection point of bisector lines of the angles of the triangle
Circumcenter is the intersection point of the sides perpendicular bisectors
Ortocenter is the intersection point of the height
Centroid is the intersection point of the medians

The average of the coordinates gives the centroid.
The other points are less trivial to find
Incenter can be found using the formula here http://www.mathopenref.com/coordincenter.html

Circumcenter P has coordinates
<br /> x_P=\frac{{x_A}^2 {x_B}-{x_A}^2 {x_C}-{x_A} {x_B}^2+{x_A} {x_C}^2-{x_A} {y_B}^2+{x_A} {y_C}^2+{x_B}^2 {x_C}-{x_B} {x_C}^2+{x_B} {y_A}^2-{x_B} {y_C}^2-{x_C} {y_A}^2+{x_C} {y_B}^2}{2 (-{x_A} {y_B}+{x_A} {y_C}+{x_B} {y_A}-{x_B} {y_C}-{x_C} {y_A}+{x_C} {y_B})}<br /> <br />

<br /> y_P=<br /> \frac{{x_A}^2 ({x_C}-{x_B})+{x_A} \left({x_B}^2-{x_C}^2+{y_B}^2-{y_C}^2\right)-{x_B}^2 {x_C}+{x_B} \left({x_C}^2-{y_A}^2+{y_C}^2\right)+{x_C} ({y_A}-{y_B}) ({y_A}+{y_B})}{2 ({x_A} ({y_B}-{y_C})+{x_B} ({y_C}-{y_A})+{x_C} ({y_A}-{y_B}))}<br />

Orthocenter O has coordinates

<br /> x_O=\frac{{x_A} ({x_B} ({y_B}-{y_A})+{x_C} ({y_A}-{y_C}))-({y_B}-{y_C}) ({x_B} {x_C}+({y_A}-{y_B}) ({y_A}-{y_C}))}{{x_A} ({y_B}-{y_C})+{x_B} ({y_C}-{y_A})+{x_C} ({y_A}-{y_B})}<br />

<br /> y_O=\frac{{y_C} (-{x_A} {y_A}+{x_B} {y_B}+{x_C} ({y_A}-{y_B}))+({x_A}-{x_B}) (({x_A}-{x_C}) ({x_B}-{x_C})+{y_A} {y_B})}{{x_A} ({y_B}-{y_C})+{x_B} ({y_C}-{y_A})+{x_C} ({y_A}-{y_B})}<br />

It can be quite hard to work it out...

Graphically it can be seen as in the following picture
4h7adu.jpg
 
Last edited:
  • #13
I am referring to my first picture where I have evaluated locus of incentre.
 
  • #14
Picture in #5
 
  • #15
Thanks Raffaelel! I have calculated incentre locus in attached image in #5.Need to get it verified.
 
Last edited:
Back
Top