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### #Actualtaby

Posted 21 May 2012 - 01:57 PM

Indeed, as other posters said the exact intersection between two lines is very easy (simple algebra), but due to floating point errors etc two lines in 3D are very unlikely to intersect exactly.

That's true regarding floating point precision, and thanks for adding that, because it is technically important. That said, it could go both ways, because sometimes there may be non-intersecting lines that are erroneously found to be intersecting because the non-zero shortest distance between them was whittled down to zero due to limited precision. What I'm saying is that you bring up a good point, and I thank you for that.

I should be more specific a lot of times: I was more referring to how the parallel postulate doesn't really apply in 3D as it does in 2D. Most cases in 3D are skew lines, where the lines are not parallel but yet still fail to intersect. It's the extra, third spatial dimension that makes the critical difference here. Now, two non-parallel planes in 3D do intersect, and it's pretty much because the extra, third spatial dimension is accompanied by the extra, second dimension inherent to the primitive in question (ie. the plane is 2D, not 1D like the line is). I honestly wish I knew the name of the theorem / postulate / axiom behind this, but it's escaping me. Maybe it's just called the generalized parallel postulate for Euclidean space, or something, I dunno. I guess the general rule of thumb is: For a pair of nD "plane" primitives, you need more than 2n embedding dimensions for the primitives to be able to be skew. So, for a pair of 1D lines you'd need a 3D space for them to be skew, and for a pair of 2D planes you'd need a 5D space for them to be skew, and for a pair of 3D hyperplanes you'd need a 7D space for them to be skew, etc, etc.

Sorry, just going around clarifying stuff that I wrote earlier. It was my fault that I didn't do so earlier.

### #26taby

Posted 21 May 2012 - 01:47 PM

Indeed, as other posters said the exact intersection between two lines is very easy (simple algebra), but due to floating point errors etc two lines in 3D are very unlikely to intersect exactly.

That's true regarding floating point precision, and thanks for adding that, because it is technically important. That said, it could go both ways, because sometimes there may be non-intersecting lines that are erroneously found to be intersecting because the non-zero shortest distance between them was whittled down to zero due to limited precision. What I'm saying is that you bring up a good point, and I thank you for that.

I should be more specific a lot of times: I was more referring to how the parallel postulate doesn't really apply in 3D as it does in 2D. Most cases in 3D are skew lines, where the lines are not parallel but yet still fail to intersect. It's the extra, third spatial dimension that makes the critical difference here. Now, two non-parallel planes in 3D do intersect, and it's pretty much because the extra, third spatial dimension is accompanied by the extra, second dimension inherent to the primitive in question (ie. the plane is 2D, not 1D like the line is). I honestly wish I knew the name of the theorem / postulate / axiom behind this, but it's escaping me. Maybe it's just called the generalized parallel postulate for Euclidean space, or something, I dunno. I guess the general rule of thumb is: For a pair of nD primitives, you need more than 2n embedding dimensions for the primitives to be able to be skew. So, for a pair of 1D lines you'd need a 3D space for them to be skew, and for a pair of 2D planes you'd need a 5D space for them to be skew, and for a pair of 3D hyperplanes you'd need a 7D space for them to be skew, etc, etc.

Sorry, just going around clarifying stuff that I wrote earlier. It was my fault that I didn't do so earlier.

### #25taby

Posted 21 May 2012 - 01:47 PM

Indeed, as other posters said the exact intersection between two lines is very easy (simple algebra), but due to floating point errors etc two lines in 3D are very unlikely to intersect exactly.

That's true regarding floating point precision, and thanks for adding that, because it is technically important. That said, it could go both ways, because sometimes there may be non-intersecting lines that are erroneously found to be intersecting because the non-zero shortest distance between them was whittled down to zero due to limited precision. What I'm saying is that you bring up a good point, and I thank you for that.

I should be more specific a lot of times: I was more referring to how the parallel postulate doesn't really apply in 3D as it does in 2D. Most cases in 3D are skew lines, where the lines are not parallel but yet still fail to intersect. It's the extra, third spatial dimension that makes the critical difference here. Now, two non-parallel planes in 3D do intersect, and it's pretty much because the extra, third spatial dimension is accompanied by the extra, second dimension inherent to the primitive in question (ie. the plane is 2D, not 1D like the line is). I honestly wish I knew the name of the theorem / postulate / axiom behind this, but it's escaping me. Maybe it's just called the generalized parallel postulate for Euclidean space, or something, I dunno. I guess the general rule of thumb is: For a pair of nD primitives, you need more than 2n embedding dimensions for the primitives to be able to be skew. So, for 1D lines you'd need a 3D space for them to be skew, and for 2D planes you'd need a 5D space for them to be skew, and for 3D hyperplanes you'd need a 7D space for them to be skew, etc, etc.

Sorry, just going around clarifying stuff that I wrote earlier. It was my fault that I didn't do so earlier.

### #24taby

Posted 21 May 2012 - 01:46 PM

Indeed, as other posters said the exact intersection between two lines is very easy (simple algebra), but due to floating point errors etc two lines in 3D are very unlikely to intersect exactly.

That's true regarding floating point precision, and thanks for adding that, because it is technically important. That said, it could go both ways, because sometimes there may be non-intersecting lines that are erroneously found to be intersecting because the non-zero shortest distance between them was whittled down to zero due to limited precision. What I'm saying is that you bring up a good point, and I thank you for that.

I should be more specific a lot of times: I was more referring to how the parallel postulate doesn't really apply in 3D as it does in 2D. Most cases in 3D are skew lines, where the lines are not parallel but yet still fail to intersect. It's the extra, third spatial dimension that makes the critical difference here. Now, two non-parallel planes in 3D do intersect, and it's pretty much because the extra, third spatial dimension is accompanied by the extra, second dimension inherent to the primitive in question (ie. the plane is 2D, not 1D like the line is). I honestly wish I knew the name of the theorem / postulate / axiom behind this, but it's escaping me. Maybe it's just called the generalized parallel postulate for Euclidean space, or something, I dunno. I guess the general rule of thumb is: For a pair of nD primitives, you need more than 2n embedding dimensions for the primitives to be able to be skew. So, for 1D lines you'd need a 3D space for them to be skew, and for 2D planes you'd need a 5D space for them to be skew, etc, etc.

Sorry, just going around clarifying stuff that I wrote earlier. It was my fault that I didn't do so earlier.

### #23taby

Posted 21 May 2012 - 01:45 PM

Indeed, as other posters said the exact intersection between two lines is very easy (simple algebra), but due to floating point errors etc two lines in 3D are very unlikely to intersect exactly.

That's true regarding floating point precision, and thanks for adding that, because it is technically important. That said, it could go both ways, because sometimes there may be non-intersecting lines that are erroneously found to be intersecting because the non-zero shortest distance between them was whittled down to zero due to limited precision. What I'm saying is that you bring up a good point, and I thank you for that.

I should be more specific a lot of times: I was more referring to how the parallel postulate doesn't really apply in 3D as it does in 2D. Most cases in 3D are skew lines, where the lines are not parallel but yet still fail to intersect. It's the extra, third spatial dimension that makes the critical difference here. Now, two non-parallel planes in 3D do intersect, and it's pretty much because the extra, third spatial dimension is accompanied by the extra, second dimension inherent to the primitive in question (ie. the plane is 2D, not 1D like the line is). I honestly wish I knew the name of the theorem / postulate / axiom behind this, but it's escaping me. Maybe it's just called the generalized parallel postulate for Euclidean space, or something, I dunno. I guess the general rule of thumb is: For a pair of nD primitives, you need more than 2n embedding dimensions for the primitives to be able to be skew. So, for 2D planes, you'd need a 5D space for them to be skew.

Sorry, just going around clarifying stuff that I wrote earlier. It was my fault that I didn't do so earlier.

### #22taby

Posted 21 May 2012 - 01:45 PM

Indeed, as other posters said the exact intersection between two lines is very easy (simple algebra), but due to floating point errors etc two lines in 3D are very unlikely to intersect exactly.

That's true regarding floating point precision, and thanks for adding that, because it is technically important. That said, it could go both ways, because sometimes there may be non-intersecting lines that are erroneously found to be intersecting because the non-zero shortest distance between them was whittled down to zero due to limited precision. What I'm saying is that you bring up a good point, and I thank you for that.

I should be more specific a lot of times: I was more referring to how the parallel postulate doesn't really apply in 3D as it does in 2D. Most cases in 3D are skew lines, where the lines are not parallel but yet still fail to intersect. It's the extra, third spatial dimension that makes the critical difference here. Now, two non-parallel planes in 3D do intersect, and it's pretty much because the extra, third spatial dimension is accompanied by the extra, second dimension inherent to the primitive in question (ie. the plane is 2D, not 1D like the line is). I honestly wish I knew the name of the theorem / postulate / axiom behind this, but it's escaping me. Maybe it's just called the generalized parallel postulate for Euclidean space, or something, I dunno. I guess the general rule of thumb is: For nD primitives, you need more than 2n embedding dimensions for the primitives to be able to be skew. So, for 2D planes, you'd need a 5D space for them to be skew.

Sorry, just going around clarifying stuff that I wrote earlier. It was my fault that I didn't do so earlier.

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