# Describing an object made by the intersection of 2 surfaces (1 Viewer)

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#### EEristavi

1. The problem statement, all variables and given/known data
Describe and sketch the geometric objects represented by the
systems of equations

2. Relevant equations
x2 + y2 + z2 = 4
x + y + z = 1

3. The attempt at a solution
I can sketch both objects:
1) sphere with center (0,0,0) and radius 2
2) "simple" plane with intersection points (on xyz plane): (1,0,0), (0,1,0), (0,0,1)

I see that their intersection will give us the circle.
However, I cant guess the center and the radius.

what I tried on this issue so far:
from eq. 2 -> z = 1 - (x +y) and put it in eq.1
we get (after few steps):
x2 + y2 + xy - x - y = 3/2
It's really hard to guess all the mentioned parameters, considering that it doesn't have the circle equation form: x2 + y2 = k (which is understandable, cause it on a "tilted" plane)

Any ideas, how to guess the mentioned variables, or change the equation in to "better" form?

#### epenguin

Homework Helper
You can't see. Don't worry. One strategy (a 'Polya Principle) try solve a simpler related problem. E.g. here the 2-d version of this 3-d problem.

#### EEristavi

E.g. here the 2-d version of this 3-d problem.
Couldn't get what you've said here

#### epenguin

Homework Helper
Couldn't get what you've said here
Instead of sphere and plane try first to solve it for circle and line, hopefully that will give you a guide. This is a fairly general strategy recommended by Polya in his short, cheap, useful, best selling book "How to solve it".

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#### WWGD

Gold Member
One way may be to rotate the plane so that it lies in the equator of the sphere, find its center and radius and then rotate back and finding the image of the center 8nder the inverse map. I think that should work.

#### Nik_2213

Slightly off-topic, but the thread title had me confused. you meant 'surfaces', wrote 'planes', for which the reply was 'straight line'...

#### EEristavi

Slightly off-topic, but the thread title had me confused. you meant 'surfaces', wrote 'planes', for which the reply was 'straight line'...
Maybe you're right. However, I think one can consider sphere surface as a plane

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#### ehild

Homework Helper
1. The problem statement, all variables and given/known data
Describe and sketch the geometric objects represented by the
systems of equations

I can sketch both objects:
1) sphere with center (0,0,0) and radius 2
2) "simple" plane with intersection points (on xyz plane): (1,0,0), (0,1,0), (0,0,1)

I see that their intersection will give us the circle.
However, I cant guess the center and the radius.

Any ideas, how to guess the mentioned variables, or change the equation in to "better" form?
You can find the radius and centre by simple Geometry. It is clear that the circle is in the x+y+z=1 plane, and its centre C is on the line (1,1,1), at the centre of the yellow triangle.

You can derive h, the distance of C from the origin O, and also the coordinates of C.
The plane extends to the big sphere of radius R=2 as shown. The blue line is the side view of the circle. What is r, the radius of the circle then?

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#### LCKurtz

Homework Helper
Gold Member
I had a bit of time to kill this morning, so here's a picture that shows what it looks like looking straight down from the first octant to the slanted plane. Only the part of the plane in the first octant is shown, and when the plane is extended it cuts the sphere in the red circle. The numbers I used to do the plots were gotten from ehild's suggestion.

I have edited to get a more accurate picture than previously posted.
Here's another shot with a different color scheme and an angle closer to the edge view of the plane:

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#### ehild

Homework Helper
I have edited to get a more accurate picture than previously posted.
Here's another shot with a different color scheme and an angle closer to the edge view of the plane:
View attachment 240335
Very suggestive picture!

#### SammyS

Staff Emeritus
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I had a bit of time to kill this morning, so here's a picture that shows what it looks like looking straight down from the first octant to the slanted plane. Only the part of the plane in the first octant is shown, and when the plane is extended it cuts the sphere in the red circle. The numbers I used to do the plots were gotten from ehild's suggestion.
[ ATTACH=full]240309[/ATTACH]
I have edited to get a more accurate picture than previously posted.
Here's another shot with a different color scheme and an angle closer to the edge view of the plane:
[ ATTACH=full]240335[/ATTACH]
Wow !

Those are helpful images, as @ehild said. I especially like the second one - which you added a little later.

@LCKurtz , I know you've been asked this some time ago, but what application/program/utility do you use to make such wonderful graphics?

#### LCKurtz

Homework Helper
Gold Member
Wow !

Those are helpful images, as @ehild said. I especially like the second one - which you added a little later.

@LCKurtz , I know you've been asked this some time ago, but what application/program/utility do you use to make such wonderful graphics?
I use Maple 13, which is probably an old version by now.

#### WWGD

Gold Member
Sorry if I missed something, but is the question about showing that the intersection is a circle, of providing the coordinates of the figure that results from the intersection?

#### Ray Vickson

Homework Helper
Sorry if I missed something, but is the question about showing that the intersection is a circle, of providing the coordinates of the figure that results from the intersection?
The OP claims the intersection is a circle, but he/she does not know how to get the center and the radius.

#### LCKurtz

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Gold Member
I'm guessing the OP figured out the answers he needed using ehild's hints, which is likely why he hasn't returned. The rest of us are just noodling around here.

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#### EEristavi

Thanks Everyone, Really nice pictures!

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