Talk:2509: Useful Geometry Formulas - Revision history

172.70.254.113 at 17:38, 24 May 2022

2022-05-24T17:38:21Z

172.70.142.239: Question re cross section

2021-09-07T11:00:38Z

Question re cross section

162.158.91.184 at 07:04, 5 September 2021

2021-09-05T07:04:04Z

GcGYSF(asterisk)P(vertical line)e at 20:28, 4 September 2021

2021-09-04T20:28:39Z

172.70.143.22 at 05:38, 3 September 2021

2021-09-03T05:38:54Z

172.69.90.75: More details on upper-right

2021-09-02T16:48:49Z

More details on upper-right

Arl guy at 14:27, 2 September 2021

2021-09-02T14:27:32Z

Arl guy at 14:20, 2 September 2021

2021-09-02T14:20:33Z

172.70.34.91 at 09:00, 2 September 2021

2021-09-02T09:00:19Z

162.158.155.247 at 08:56, 2 September 2021

2021-09-02T08:56:59Z

← Older revision		Revision as of 17:38, 24 May 2022
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	'''Cross-section - reword please''' I made a wording error in describing all points on the "cross-section" of the sphere as being equidistant from the centre, and I'm stuck on how to fix it concisely. Problem is that "cross-section" is ambiguous - I think mathematically it ''could'' mean the circle at the intersection of the hollow sphere and the plane, but to me it usually means the part of the plane where it intersects the solid sphere, which makes the statement false. I want to replace it with "circle", with the quote marks, but we're actually talking about the ellipse-cum-circle so I'm not sure if that's good enough. [[Special:Contributions/172.70.142.239\|172.70.142.239]] 11:00, 7 September 2021 (UTC)		'''Cross-section - reword please''' I made a wording error in describing all points on the "cross-section" of the sphere as being equidistant from the centre, and I'm stuck on how to fix it concisely. Problem is that "cross-section" is ambiguous - I think mathematically it ''could'' mean the circle at the intersection of the hollow sphere and the plane, but to me it usually means the part of the plane where it intersects the solid sphere, which makes the statement false. I want to replace it with "circle", with the quote marks, but we're actually talking about the ellipse-cum-circle so I'm not sure if that's good enough. [[Special:Contributions/172.70.142.239\|172.70.142.239]] 11:00, 7 September 2021 (UTC)
		+
		+	Can somebody draw what these 2d shapes are supposed to look like?

← Older revision		Revision as of 11:00, 7 September 2021
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	'''Subtext??:''' I wonder if this comic implies that we are being transformed from 3d creatures into 2D creatures...		'''Subtext??:''' I wonder if this comic implies that we are being transformed from 3d creatures into 2D creatures...
	hadaso --[[Special:Contributions/162.158.91.184\|162.158.91.184]] 07:04, 5 September 2021 (UTC)		hadaso --[[Special:Contributions/162.158.91.184\|162.158.91.184]] 07:04, 5 September 2021 (UTC)
		+
		+	'''Cross-section - reword please''' I made a wording error in describing all points on the "cross-section" of the sphere as being equidistant from the centre, and I'm stuck on how to fix it concisely. Problem is that "cross-section" is ambiguous - I think mathematically it ''could'' mean the circle at the intersection of the hollow sphere and the plane, but to me it usually means the part of the plane where it intersects the solid sphere, which makes the statement false. I want to replace it with "circle", with the quote marks, but we're actually talking about the ellipse-cum-circle so I'm not sure if that's good enough. [[Special:Contributions/172.70.142.239\|172.70.142.239]] 11:00, 7 September 2021 (UTC)

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 One aspect of this comic that has not been mentioned is how strong the dashed line convention is if you are very familiar with these sorts of representations. I found it almost impossible to force my brain to see this as a 2D diagram with solid and (superflous) dashed lines. That third dimension just keeps popping out, although I could occasionally reverse the convention and see the dashed lines as in front rather than in back. [[User:Arl guy|Arl guy]] ([[User talk:Arl guy|talk]]) 14:27, 2 September 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)
+Can 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)
 : ''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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 :Because of dependencies, as the theta moves the other similar angles ''might'' be thetas too, or they could be kept as RAs and (?)three ''other'' chosen angles could be distorted (in or out of the page, by a calculable and entirely derivative amount) to compensate. Or, like the Triangle illusion, there's overlaid duplicate edges with not all/any vertices actually being the same, just in the same illustrated place. (Again, making a mockery of the 'simple' diagram, but we're already way past that. ;) ) [[Special:Contributions/162.158.155.247|162.158.155.247]] 08:56, 2 September 2021 (UTC)
 ::Sure, but we were talking about the 2D figure and taking it at "face" value - the internal angle is defined while the ones you'd actually use for the parallelogram area calculations aren't, and neither are the dimensions of the upper rectangle. In a proper geometric figure you'd have markings showing which lengths are equal and which angles are right angles.[[Special:Contributions/172.70.143.22|172.70.143.22]] 05:38, 3 September 2021 (UTC)

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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)
 :::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)
 : 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)

← Older revision		Revision as of 05:38, 3 September 2021
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	:There's always many possible {{w\|Ames room\|unindicated perspective tricks}} (or axonometric/othographic ones, as might be more appropriate) in which the components of lines and relationships towards/away from the viewer are non-zero (or {{w\|Penrose triangle\|not non-zero in the way expected}}). Various of the shapes involved could be infinitely warped with leans, curves or even highly inflected wiggles to allow ''some'' 'as expected' profile even as others that should be connected are not. (Though it would make the surface 'planes' warped and distorted, even more against the conventions of wireframe diagrams.)		:There's always many possible {{w\|Ames room\|unindicated perspective tricks}} (or axonometric/othographic ones, as might be more appropriate) in which the components of lines and relationships towards/away from the viewer are non-zero (or {{w\|Penrose triangle\|not non-zero in the way expected}}). Various of the shapes involved could be infinitely warped with leans, curves or even highly inflected wiggles to allow ''some'' 'as expected' profile even as others that should be connected are not. (Though it would make the surface 'planes' warped and distorted, even more against the conventions of wireframe diagrams.)
	:Because of dependencies, as the theta moves the other similar angles ''might'' be thetas too, or they could be kept as RAs and (?)three ''other'' chosen angles could be distorted (in or out of the page, by a calculable and entirely derivative amount) to compensate. Or, like the Triangle illusion, there's overlaid duplicate edges with not all/any vertices actually being the same, just in the same illustrated place. (Again, making a mockery of the 'simple' diagram, but we're already way past that. ;) ) [[Special:Contributions/162.158.155.247\|162.158.155.247]] 08:56, 2 September 2021 (UTC)		:Because of dependencies, as the theta moves the other similar angles ''might'' be thetas too, or they could be kept as RAs and (?)three ''other'' chosen angles could be distorted (in or out of the page, by a calculable and entirely derivative amount) to compensate. Or, like the Triangle illusion, there's overlaid duplicate edges with not all/any vertices actually being the same, just in the same illustrated place. (Again, making a mockery of the 'simple' diagram, but we're already way past that. ;) ) [[Special:Contributions/162.158.155.247\|162.158.155.247]] 08:56, 2 September 2021 (UTC)
		+	::Sure, but we were talking about the 2D figure and taking it at "face" value - the internal angle is defined while the ones you'd actually use for the parallelogram area calculations aren't, and neither are the dimensions of the upper rectangle. In a proper geometric figure you'd have markings showing which lengths are equal and which angles are right angles.[[Special:Contributions/172.70.143.22\|172.70.143.22]] 05:38, 3 September 2021 (UTC)

@@ Line 60: / Line 60: @@
 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)
 ; It's 3am (okay 5am) and I made it really long!

← Older revision		Revision as of 14:27, 2 September 2021
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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)
		+
		+	;Can you "see" this as 2D?
		+	One aspect of this comic that has not been mentioned is how strong the dashed line convention is if you are very familiar with these sorts of representations. I found it almost impossible to force my brain to see this as a 2D diagram with solid and (superflous) dashed lines. That third dimension just keeps popping out, although I could occasionally reverse the convention and see the dashed lines as in front rather than in back. [[User:Arl guy\|Arl guy]] ([[User talk:Arl guy\|talk]]) 14:27, 2 September 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)		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)

@@ Line 73: / Line 73: @@
 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 assume that the two rectangles have equal width and height, then it can be mathematically proven that the angles must be equal (probably using congruent triangles). However this assumption is not stated on the figure. That said, you would make the same assumption for the 3D figure, along with a whole bunch of other assumptions of course.[[Special:Contributions/172.70.147.23|172.70.147.23]] 06:31, 2 September 2021 (UTC) Edit: The two rectangles must have equal width and height to make the rest of the shapes parallelograms in the first place. If they aren't identical you get trapezoids all round and possibly a bunch of different angles. [[Special:Contributions/172.70.143.22|172.70.143.22]] 06:43, 2 September 2021 (UTC)
 :There's always many possible {{w|Ames room|unindicated perspective tricks}} (or axonometric/othographic ones, as might be more appropriate) in which the components of lines and relationships towards/away from the viewer are non-zero (or {{w|Penrose triangle|not non-zero in the way expected}}). Various of the shapes involved could be infinitely warped with leans, curves or even highly inflected wiggles to allow ''some'' 'as expected' profile even as others that should be connected are not. (Though it would make the surface 'planes' warped and distorted, even more against the conventions of wireframe diagrams.)
 :Because of dependencies, as the theta moves the other similar angles ''might'' be thetas too, or they could be kept as RAs and (?)three ''other'' chosen angles could be distorted (in or out of the page, by a calculable and entirely derivative amount) to compensate. Or, like the Triangle illusion, there's overlaid duplicate edges with not all/any vertices actually being the same, just in the same illustrated place. (Again, making a mockery of the 'simple' diagram, but we're already way past that. ;) ) [[Special:Contributions/162.158.155.247|162.158.155.247]] 08:56, 2 September 2021 (UTC)

← Older revision		Revision as of 08:56, 2 September 2021
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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)
	:If you assume that the two rectangles have equal width and height, then it can be mathematically proven that the angles must be equal (probably using congruent triangles). However this assumption is not stated on the figure. That said, you would make the same assumption for the 3D figure, along with a whole bunch of other assumptions of course.[[Special:Contributions/172.70.147.23\|172.70.147.23]] 06:31, 2 September 2021 (UTC) Edit: The two rectangles must have equal width and height to make the rest of the shapes parallelograms in the first place. If they aren't identical you get trapezoids all round and possibly a bunch of different angles. [[Special:Contributions/172.70.143.22\|172.70.143.22]] 06:43, 2 September 2021 (UTC)		:If you assume that the two rectangles have equal width and height, then it can be mathematically proven that the angles must be equal (probably using congruent triangles). However this assumption is not stated on the figure. That said, you would make the same assumption for the 3D figure, along with a whole bunch of other assumptions of course.[[Special:Contributions/172.70.147.23\|172.70.147.23]] 06:31, 2 September 2021 (UTC) Edit: The two rectangles must have equal width and height to make the rest of the shapes parallelograms in the first place. If they aren't identical you get trapezoids all round and possibly a bunch of different angles. [[Special:Contributions/172.70.143.22\|172.70.143.22]] 06:43, 2 September 2021 (UTC)
		+	:There's always many possible {{w\|Ames room\|unindicated perspective tricks}} (or axonometric/othographic ones, as might be more appropriate) in which the components of lines and relationships towards/away from the viewer are non-zero (or {{w\|Penrose triangle\|not non-zero in the way expected}}). Various of the shapes involved could be infinitely warped with leans, curves or even highly inflected wiggles to allow ''some'' 'as expected' profile even as others that should be connected are not. (Though it would make the surface 'planes' warped and distorted, even more against the conventions of wireframe diagrams.)
		+	:Because of dependencies, as the theta moves the other similar angles ''might'' be thetas too, or they could be kept as RAs and (?)three ''other'' angles could be distorted (in or out of the page) to compensate. Or, like the Triangle illusion, there's overlaid duplicate edges with not all/any vertices actually being the same, just in the same illustrated place. (Again, making a mockery of the 'simple' diagram, but we're already way past that. ;) ) [[Special:Contributions/162.158.155.247\|162.158.155.247]] 08:56, 2 September 2021 (UTC)