Describing an object made by the intersection of 2 surfaces

[EEristavi]

Homework Statement

Describe and sketch the geometric objects represented by the
systems of equations

Homework Equations

x² + y² + z² = 4
x + y + z = 1

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 into "better" form?

 

[epenguin]

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]

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".

 

[WWGD]

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

 

[ehild]

However, I think one can consider sphere surface as a plane

No. Plane is a surface, sphere is a surface, but sphere is not plane.
In Mathematics, plane is a flat or level surface that continues in all directions.

 

[ehild]

Homework Statement

Describe and sketch the geometric objects represented by the
systems of equationsI 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 into "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?

 

[LCKurtz]

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:

 

[ehild]

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]

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]

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]

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]

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]

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.

 

[SammyS]

I use Maple 13, which is probably an old version by now.

Thanks.

 

[EEristavi]

Thanks Everyone, Really nice pictures!