Question about general equation for a line in 3d [modified]
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Someone was asking about that a while back: Why is there no scalar equation for a line in three dimensions ? http://www.physicsforums.com/showthread.php?t=165472[^] I think the R-difference (from R3 to R1) in dimensionality accounts for it. That's why there's no form analogous to Ax + By + C that includes a z-coefficient. Any ideas ? The only equation that might come close to fitting the bill is this one: Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 If B2 - 4AC is zero this equation might represent a line, or two lines if less than zero. But this requires a section by a plane of a conic surface; and although it's a scalar equation, it still isn't linear; so I don't think it's suitable.
modified on Monday, March 14, 2011 5:09 PM
In 2D a line could be represented as
Ax+By+C=0; this merely is a single constraint, reducing the 2D space to a one-dimensional collection of points. Similarly in 3D a plane could beAx+By+Cz+D=0, and a line would be the intersection of two planes (it takes two constraints to reduce a 3D space to a one-dimensional object). Another approach is using a parameter, say t. Then a line would be the combination of:x=At+B, andy=Ct+D, andz=Et+F(take as many as you need for an N-dimensional space). One can think of t as time, and the point traveling along the intended line over time. :)Luc Pattyn [Forum Guidelines] [My Articles] Nil Volentibus Arduum
Please use <PRE> tags for code snippets, they preserve indentation, improve readability, and make me actually look at the code.
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So, a 3D version of y = mx + b?
y = mx + Mz + b
Note that m and M are just constants that may not have the same meaning as m in the original equation. Also note that I didn't understand a word you said, so I could be completely off base.
Nope, that would be a plane, as for every pair (x,z) you get a y-value. :)
Luc Pattyn [Forum Guidelines] [My Articles] Nil Volentibus Arduum
Please use <PRE> tags for code snippets, they preserve indentation, improve readability, and make me actually look at the code.
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whoooooooosh![^] [sound of that rapidly flying right over my head, in case youtube not working for you!]
Dave Find Me On: Web|Facebook|Twitter|LinkedIn
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landing gear inspection? :)
Luc Pattyn [Forum Guidelines] [My Articles] Nil Volentibus Arduum
Please use <PRE> tags for code snippets, they preserve indentation, improve readability, and make me actually look at the code.
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landing gear inspection? :)
Luc Pattyn [Forum Guidelines] [My Articles] Nil Volentibus Arduum
Please use <PRE> tags for code snippets, they preserve indentation, improve readability, and make me actually look at the code.
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so thats not the plane he's on about travelling from point a to b to c then.......... :rolleyes:
Dave Find Me On: Web|Facebook|Twitter|LinkedIn
Folding Stats: Team CodeProject
No, mathematical planes are infinite; maybe slower than yours, but a lot bigger. :)
Luc Pattyn [Forum Guidelines] [My Articles] Nil Volentibus Arduum
Please use <PRE> tags for code snippets, they preserve indentation, improve readability, and make me actually look at the code.
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Someone was asking about that a while back: Why is there no scalar equation for a line in three dimensions ? http://www.physicsforums.com/showthread.php?t=165472[^] I think the R-difference (from R3 to R1) in dimensionality accounts for it. That's why there's no form analogous to Ax + By + C that includes a z-coefficient. Any ideas ? The only equation that might come close to fitting the bill is this one: Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 If B2 - 4AC is zero this equation might represent a line, or two lines if less than zero. But this requires a section by a plane of a conic surface; and although it's a scalar equation, it still isn't linear; so I don't think it's suitable.
modified on Monday, March 14, 2011 5:09 PM
GAMerritt wrote:
Why is there no scalar equation for a line in three dimensions ?
I'm thinking you have the question back to front. Let's pretend a 1D universe. Then you have this equation:
ax = d
aka, in a 1D world, a scalar equation defines a point, or 0D object. In a 2D universe, you get this equation:
ax + by = d
aka, in a 2D world, a scalar equation defines a line, or 1D object. In a 3D universe, you get this equation:
ax + by + cz= d
aka, in a 3D world, a scalar equation defines a plane, or 2D object. To extrapolate... Σ(1->n) coeff [n] * axis [n] = d give an n-1 dimensional object. We just happen to have names for the previous things. I have no idea what you'd call a 3d scalar object in a 4d world. A hyper-plane? I hope that made some sense. Knowing how these things work, some italian mathematician will have written this all much more nicely and elegantly four hundred years ago. Iain. [Modified to fix typo - twice]
I am one of "those foreigners coming over here and stealing our jobs". Yay me!
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GAMerritt wrote:
Why is there no scalar equation for a line in three dimensions ?
I'm thinking you have the question back to front. Let's pretend a 1D universe. Then you have this equation:
ax = d
aka, in a 1D world, a scalar equation defines a point, or 0D object. In a 2D universe, you get this equation:
ax + by = d
aka, in a 2D world, a scalar equation defines a line, or 1D object. In a 3D universe, you get this equation:
ax + by + cz= d
aka, in a 3D world, a scalar equation defines a plane, or 2D object. To extrapolate... Σ(1->n) coeff [n] * axis [n] = d give an n-1 dimensional object. We just happen to have names for the previous things. I have no idea what you'd call a 3d scalar object in a 4d world. A hyper-plane? I hope that made some sense. Knowing how these things work, some italian mathematician will have written this all much more nicely and elegantly four hundred years ago. Iain. [Modified to fix typo - twice]
I am one of "those foreigners coming over here and stealing our jobs". Yay me!
Iain Clarke, Warrior Programmer wrote:
ax + by + cy= d
did you mean ax + by + cz = d?
"If your actions inspire others to dream more, learn more, do more and become more, you are a leader." - John Quincy Adams "Let me get this straight. You know her. She knows you. But she wants to eat him. And everybody's okay with this?" - Timon
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GAMerritt wrote:
Why is there no scalar equation for a line in three dimensions ?
I'm thinking you have the question back to front. Let's pretend a 1D universe. Then you have this equation:
ax = d
aka, in a 1D world, a scalar equation defines a point, or 0D object. In a 2D universe, you get this equation:
ax + by = d
aka, in a 2D world, a scalar equation defines a line, or 1D object. In a 3D universe, you get this equation:
ax + by + cz= d
aka, in a 3D world, a scalar equation defines a plane, or 2D object. To extrapolate... Σ(1->n) coeff [n] * axis [n] = d give an n-1 dimensional object. We just happen to have names for the previous things. I have no idea what you'd call a 3d scalar object in a 4d world. A hyper-plane? I hope that made some sense. Knowing how these things work, some italian mathematician will have written this all much more nicely and elegantly four hundred years ago. Iain. [Modified to fix typo - twice]
I am one of "those foreigners coming over here and stealing our jobs". Yay me!
Iain Clarke, Warrior Programmer wrote:
ax + by + cyz= d
I suppose the number of dimensions is proportionally direct to the number of variables in a scalar equation XD
Iain Clarke, Warrior Programmer wrote:
I am one of "those foreigners coming over here and stealing our jobs". Yay me!
Then we are two! :-D
"Whether you think you can, or you think you can't--either way, you are right." — Henry Ford
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Iain Clarke, Warrior Programmer wrote:
ax + by + cy= d
did you mean ax + by + cz = d?
"If your actions inspire others to dream more, learn more, do more and become more, you are a leader." - John Quincy Adams "Let me get this straight. You know her. She knows you. But she wants to eat him. And everybody's okay with this?" - Timon
ahmed zahmed wrote:
did you mean ax + by + cz = d?
It is possible, I suppose...
I am one of "those foreigners coming over here and stealing our jobs". Yay me!
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GAMerritt wrote:
Why is there no scalar equation for a line in three dimensions ?
I'm thinking you have the question back to front. Let's pretend a 1D universe. Then you have this equation:
ax = d
aka, in a 1D world, a scalar equation defines a point, or 0D object. In a 2D universe, you get this equation:
ax + by = d
aka, in a 2D world, a scalar equation defines a line, or 1D object. In a 3D universe, you get this equation:
ax + by + cz= d
aka, in a 3D world, a scalar equation defines a plane, or 2D object. To extrapolate... Σ(1->n) coeff [n] * axis [n] = d give an n-1 dimensional object. We just happen to have names for the previous things. I have no idea what you'd call a 3d scalar object in a 4d world. A hyper-plane? I hope that made some sense. Knowing how these things work, some italian mathematician will have written this all much more nicely and elegantly four hundred years ago. Iain. [Modified to fix typo - twice]
I am one of "those foreigners coming over here and stealing our jobs". Yay me!
-
Someone was asking about that a while back: Why is there no scalar equation for a line in three dimensions ? http://www.physicsforums.com/showthread.php?t=165472[^] I think the R-difference (from R3 to R1) in dimensionality accounts for it. That's why there's no form analogous to Ax + By + C that includes a z-coefficient. Any ideas ? The only equation that might come close to fitting the bill is this one: Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 If B2 - 4AC is zero this equation might represent a line, or two lines if less than zero. But this requires a section by a plane of a conic surface; and although it's a scalar equation, it still isn't linear; so I don't think it's suitable.
modified on Monday, March 14, 2011 5:09 PM
the linear equation is, in "geometric term" OA . OM = 0 where OA is your line vector and M a point in the plane in 3D I guess you can try | OA ^ OM | = 0 So if AO = (a, b, c) and OM = (x, y, z) something along the lines of (IIR, might have some sign error) | (bz - cy, cx - az, ay - bx ) | = 0 (bz - cy)^2 + (cx - az)^2 + (ay -bx)^2 = 0
A train station is where the train stops. A bus station is where the bus stops. On my desk, I have a work station.... _________________________________________________________ My programs never have bugs, they just develop random features.
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Iain Clarke, Warrior Programmer wrote:
ax + by + cz= d
That's the equation for a plane. A line in 3 dimensions is the intersection of two planes. Cheers, Drew.
You are right, I did miswrite it. I do know it's a plane - that was kind of the central thesis. Seems I should not try to be deep near bedtime! Iain.
I am one of "those foreigners coming over here and stealing our jobs". Yay me!