Editing Talk:2509: Useful Geometry Formulas
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The "decorative stripes and dotted lines" are the parts of the diagrams that are intended to indicate the third dimension. The conceit of the comic is that these are superfluous. [[User:Barmar|Barmar]] ([[User talk:Barmar|talk]]) 02:56, 31 August 2021 (UTC) | The "decorative stripes and dotted lines" are the parts of the diagrams that are intended to indicate the third dimension. The conceit of the comic is that these are superfluous. [[User:Barmar|Barmar]] ([[User talk:Barmar|talk]]) 02:56, 31 August 2021 (UTC) | ||
− | + | Ca someone explain how the last one works? [[User:GcGYSF(asterisk)P(vertical line)e|GcGYSF(asterisk)P(vertical line)e]] ([[User talk:GcGYSF(asterisk)P(vertical line)e|talk]]) 04:28, 31 August 2021 (UTC) | |
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: ''bh'' is the area of the front face. The top face is a parallelogram with sides ''d'' and ''b'', with an angle of ''θ'' between them, so its area is ''d b sin(θ)''. The right face is a parallelogram with sides ''d'' and ''h'', with an angle of ''90º - θ'' between them, so its area is ''h d sin(90º - θ) = h d cos(θ)''. So the area of the whole picture is ''bh + d b sin(θ) + d h cos(θ)''. | : ''bh'' is the area of the front face. The top face is a parallelogram with sides ''d'' and ''b'', with an angle of ''θ'' between them, so its area is ''d b sin(θ)''. The right face is a parallelogram with sides ''d'' and ''h'', with an angle of ''90º - θ'' between them, so its area is ''h d sin(90º - θ) = h d cos(θ)''. So the area of the whole picture is ''bh + d b sin(θ) + d h cos(θ)''. | ||
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::Surely ripe for a table, in place of much of the longhand paragraph spiel (which could be kept, but simpler for just the narrative but otherwise non-technical details)... "Shape (2D)", "Area", "Pretended Shape (3D)", "Surface Area", "Volume", ¿"Notes"? (Not sure about specific Notes, some things could/should be said below the formulae/descriptions in the relevent cell to which that matters, in special cases where necessary, which might be better than a Notes either empty or jammed up with all the combined row-specific corollaries, etc, that I can imagine.) Anyway, an idea. [[Special:Contributions/141.101.76.11|141.101.76.11]] 11:56, 31 August 2021 (UTC) | ::Surely ripe for a table, in place of much of the longhand paragraph spiel (which could be kept, but simpler for just the narrative but otherwise non-technical details)... "Shape (2D)", "Area", "Pretended Shape (3D)", "Surface Area", "Volume", ¿"Notes"? (Not sure about specific Notes, some things could/should be said below the formulae/descriptions in the relevent cell to which that matters, in special cases where necessary, which might be better than a Notes either empty or jammed up with all the combined row-specific corollaries, etc, that I can imagine.) Anyway, an idea. [[Special:Contributions/141.101.76.11|141.101.76.11]] 11:56, 31 August 2021 (UTC) | ||
:::I think the formulas are correct. Those given should be from the text book, not for those with ellipse bases. Someone has put a lot of work into giving these complicated formulas for the cone and cylinder. But I think that is overkill. I have added to the explanation the simple versions before, and would suggest deleting the complicated, which was never the intention of either text book or Randall! ;-)--[[User:Kynde|Kynde]] ([[User talk:Kynde|talk]]) 12:36, 31 August 2021 (UTC) | :::I think the formulas are correct. Those given should be from the text book, not for those with ellipse bases. Someone has put a lot of work into giving these complicated formulas for the cone and cylinder. But I think that is overkill. I have added to the explanation the simple versions before, and would suggest deleting the complicated, which was never the intention of either text book or Randall! ;-)--[[User:Kynde|Kynde]] ([[User talk:Kynde|talk]]) 12:36, 31 August 2021 (UTC) | ||
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: Surface area. Not volume. My bad. I usually consider volume associated with pics like like that. Don't use surface area much. [[User:Baffo32|Baffo32]] ([[User talk:Baffo32|talk]]) 22:22, 1 September 2021 (UTC) | : Surface area. Not volume. My bad. I usually consider volume associated with pics like like that. Don't use surface area much. [[User:Baffo32|Baffo32]] ([[User talk:Baffo32|talk]]) 22:22, 1 September 2021 (UTC) | ||
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Unconvinced by the cone! The equation shown, is correct for an isosceles triangle with a half-ellipse on its base. But that shape has 'corners' where the sides meet that half-ellipse. In a 3D projected view of an actual cone, the sides will meet the base ellipse at a tangent, meaning that it is more than a half-ellipse. But I suppose it's close enough as an approximation...[[Special:Contributions/172.69.55.131|172.69.55.131]] 15:57, 1 September 2021 (UTC) | Unconvinced by the cone! The equation shown, is correct for an isosceles triangle with a half-ellipse on its base. But that shape has 'corners' where the sides meet that half-ellipse. In a 3D projected view of an actual cone, the sides will meet the base ellipse at a tangent, meaning that it is more than a half-ellipse. But I suppose it's close enough as an approximation...[[Special:Contributions/172.69.55.131|172.69.55.131]] 15:57, 1 September 2021 (UTC) | ||
: I verified your claim by imagining the surface of the cone as formed by a set of lines extending from the different points on the ellipse to a single fixed point at the tip. No matter where you put that tip point, the outermost lines seem tangent to the ellipse. Seems it works for both perspective and orthographic projections. Updated the explanation. Randall's formula is incorrect, especially for very short cone projections. [[User:Baffo32|Baffo32]] ([[User talk:Baffo32|talk]]) 22:46, 1 September 2021 (UTC) | : I verified your claim by imagining the surface of the cone as formed by a set of lines extending from the different points on the ellipse to a single fixed point at the tip. No matter where you put that tip point, the outermost lines seem tangent to the ellipse. Seems it works for both perspective and orthographic projections. Updated the explanation. Randall's formula is incorrect, especially for very short cone projections. [[User:Baffo32|Baffo32]] ([[User talk:Baffo32|talk]]) 22:46, 1 September 2021 (UTC) | ||
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; It's 3am (okay 5am) and I made it really long! | ; It's 3am (okay 5am) and I made it really long! | ||
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If you don't assume that the bottom right figure is 3D, what's the justification for projecting upward and assuming that the angle theta is also the angle of the top parallelogram? [[User:Arl guy|Arl guy]] ([[User talk:Arl guy|talk]]) 02:25, 2 September 2021 (UTC) | If you don't assume that the bottom right figure is 3D, what's the justification for projecting upward and assuming that the angle theta is also the angle of the top parallelogram? [[User:Arl guy|Arl guy]] ([[User talk:Arl guy|talk]]) 02:25, 2 September 2021 (UTC) | ||
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