Difference between revisions of "3D Geometry"

 
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'''3D Geometry''' deals with objects in 3 [[dimension]]s.  For example, a drawing on a piece of paper is 2 dimensional since it has lenght and width.  But a baseball, on the other hand, is three dimensional because it not only has lenght and width but also depth.
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'''3D Geometry''' deals with objects in 3 [[dimension]]s.  For example, a drawing on a piece of paper is 2 dimensional since it has lenght and width.  But a baseball, on the other hand, is three dimensional because it not only has lenght and width, but also depth.
  
 
== Making 3D Problems 2D ==
 
== Making 3D Problems 2D ==
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First we note that the distance of a point to a line is usually meant to be the ''shortest'' distance between the point and the line.  This occurs when the perpendicular to the line segment through the point is drawn.
 
First we note that the distance of a point to a line is usually meant to be the ''shortest'' distance between the point and the line.  This occurs when the perpendicular to the line segment through the point is drawn.
  
Now we know what we are looking for, we can choose an appropriate cross-section to look at.  We choose to look at the cross-section containing <math> A, B </math> and the center of the sphere as shown in the following diagram:
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Now that we know what we are looking for, we can choose an appropriate cross-section to look at.  We choose to look at the cross-section containing <math> A, B </math> and the center of the sphere as shown in the following diagram:
  
 
<center>[[Image:sphere3d.PNG]]</center>
 
<center>[[Image:sphere3d.PNG]]</center>
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<center>[[Image:sphere3dtriangle.PNG]]</center>
 
<center>[[Image:sphere3dtriangle.PNG]]</center>
  
From here, we can note the 30-60-90 triangle or the Pythagorean Theorem to find that <math> x = \sqrt{3} </math> units.
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From here, we can note the 30-60-90 triangle, or the Pythagorean Theorem, to find that <math> x = \sqrt{3} </math> units.
  
 
== See also ==
 
== See also ==
 
* [[Geometry]]
 
* [[Geometry]]

Revision as of 17:45, 5 July 2006

3D Geometry deals with objects in 3 dimensions. For example, a drawing on a piece of paper is 2 dimensional since it has lenght and width. But a baseball, on the other hand, is three dimensional because it not only has lenght and width, but also depth.

Making 3D Problems 2D

A very common technique for approaching 3D Geometry problems is to make it 2D. We can do this by looking at certain cross-sections of the diagram one at a time.

Example

On a sphere with a radius of 2 units, the points $A$ and $B$ are 2 units away from each other. Compute the distance from the center of the sphere to the line segment $AB.$

Solution

First we note that the distance of a point to a line is usually meant to be the shortest distance between the point and the line. This occurs when the perpendicular to the line segment through the point is drawn.

Now that we know what we are looking for, we can choose an appropriate cross-section to look at. We choose to look at the cross-section containing $A, B$ and the center of the sphere as shown in the following diagram:

Sphere3d.PNG

We now draw in the perpendicular to $AB$:

Sphere3dtriangle.PNG

From here, we can note the 30-60-90 triangle, or the Pythagorean Theorem, to find that $x = \sqrt{3}$ units.

See also