Talk:2565: Latency

2022-01-08T06:09:39Z

162.158.183.232:

Ha! Welcome to my life. Just thought to check if there was a new xkcd yet (at 04:45, GMT) after spending the last five hours messing semi-manually with some geodata. Ok, the first three hours was in the text editor looking at the raw JSON file, and the next two was writing a Perl script to redo everything I had already done (and more, but not yet everything I will eventually want to do) without the fallible human element. Once the fallible human element has polished the script up to account for unforseen circumstances. [[Special:Contributions/172.70.85.73|172.70.85.73]] 04:51, 8 January 2022 (UTC)

what is SCAPDFATIAT
OH what is says in the Comic

Right, Someone Copies and Pastes From a Thing Into Another Thing [[Special:Contributions/172.70.210.183|172.70.210.183]] 05:36, 8 January 2022 (UTC)

I can relate to this. In fact, i use 2 computer screens just for that: I copy data from software ''X, screen 1'' to quickly paste it into software ''Y, screen 2''. [[Special:Contributions/162.158.183.232|162.158.183.232]] 06:09, 8 January 2022 (UTC)

2509: Useful Geometry Formulas

2021-09-01T12:53:10Z

162.158.183.232:

{{comic
| number = 2509
| date = August 30, 2021
| title = Useful Geometry Formulas
| image = useful_geometry_formulas.png
| titletext = Geometry textbooks always try to trick you by adding decorative stripes and dotted lines.
}}

==Explanation==
{{incomplete|Created by a STRIPED AND DOTTED TEXTBOOK ILLUSTRATOR. Explain the formulas for each of the areas, and also the correct formula for the 3D object they seems to represent. Do NOT delete this tag too soon.}}
This comic showcases area formulae for four two-dimensional geometric shapes which each have extra dotted and/or solid lines making them look like illustrations for 3-dimensional objects - the first, a simple equation for a circle, the second an equation for a triangle with a semi-elliptic base, the third an equation for a rectangle with an elliptical base and top, and the fourth a hexagon consisting of two opposing right angled corners and two parallel diagonal lines connecting their sides. In each case, only the outline of each shape is measured.

Such illustrations are commonly found in geometry textbooks, which need to depict three-dimensional figures on a two-dimensional page. They use slanted lines to indicate edges receding into the distance, and dashed lines to indicate an edge occluded by nearer parts of the solid. The joke is that the formulae given here are for the area of each two-dimensional shape within its outer solid lines, not for the surface area or volume of the illustrated 3D object (as would be shown in the geometry textbook). The title text continues the joke by claiming that the dotted lines are simply decorative.

The illustrations depict the following plane or solid figures, depending on the interpretation.

Top left.&emsp;A circle (illustrating a sphere) with radius r. The equation for the area of a circle is A = πr2 as is given below the figure. The surface area of a sphere is 4πr2 , which is what we would have expected from the figure. The volume of a sphere is 4/3 πr3.

Top right.&emsp;An isosceles triangle of height h combined with a semi-ellipse with semiaxes a and b (illustrating a right elliptic cone). The area of the triangle is bh, and the area of the semi-ellipse is π/2 ab. The equation for this area is A = 1/2 πab + bh as is given below the figure. However, if this was in a text book then a=b even if drawn like this, thus the cone has a circular base, in the 3D drawing. Such a "normal" cone has an area A = πb^2 + πbh. (a=b). That cone's volume would be πr^2*h/3. Taking the 3D drawing literal with a≠b then the lateral surface area of a right elliptic cone is 2a√(b2 + h2) ∫01 √(a²h²(t²-1) - b²(a²+h²t²)/a²(t²-1)(b²+h²)) dt. The volume is π/3 abh.

Bottom left.&emsp;A rectangle of width d and height h between two semi-ellipses of semi-minor axis r (illustrating a right elliptic cylinder). The area of the rectangle is dh and the area of the two half-ellipses equals the area of one full ellipse, π/2 dr. The equation for this area is A = d(πr/2 + h) as is given below the figure. For a 3D representation the cylinder has circular base so d = 2r, (not elliptical as indicated in the 2D drawing). Such a cylinder has a surface area of 2πr^2 + πdh. The volume of such a cylinder is πr^2h. Taking the 3D drawing literal with d≠2r then the lateral surface area of the right elliptic cylinder is 4h ∫01 √(1 - t²(1-4r²/d²)/1 - t²) dt. The volume is π/2 rdh.

Bottom right.&emsp;A convex hexagon with three pairs of parallel sides and two right angles at opposite vertices (illustrating a rhomboid-based prism). The area of the rectangle representing the front face of the prism is bh. The area of the upper parallelogram is db sin θ. The area of the right parallelogram is dh cos θ. The equation for this area is A = bh + d(b sinθ + h cosθ) as is given below the figure. The surface area of the prism would be 2bh + 2db sin θ + 2dh. The volume is bdh sin θ. Assuming a 3D shape, θ can be artificially altered by the projection; the assumption could be made that θ is 90 degrees, and sin θ is 1 (and therefore can be eliminated from the formulas), but since θ is marked, such an assumption might not be valid.

In the history of the development of computer-generated 3D graphics, calculations of the apparent visual area taken up by the projection of a volume may have been useful in occlusion-like optimizations, where each drawn pixel may be passed through many fragment shaders.

==Transcript==
{{incomplete transcript|Do NOT delete this tag too soon.}}
:[Four figures in two rows of two, each depicts a two-dimensional representation of a three-dimensional object, with solid lines in front and dotted lines behind. Each figure has some labeled dimensions represented with arrows and a formula underneath indicating its area. Above the four figures is a header:]
:Useful geometry formulas

:[Top left; a 'sphere', or a circle with a concentrict half-dotted ellipse sharing its major axis, with the shared semi-major radius labeled 'r']
:A = πr²

:[Top right; a 'cone', or a triangle with the base replaced by a half-dotted ellipse. The triangular/conic height is 'h'. The ellipse in place of the base has semi-minor axis 'a' and major axis 'b']
:A = 1/2 πab + bh

:[Bottom left; a 'cylinder', or a pair of ellipses connected by verticals. The vertical side/edge is shon as height 'h'. The ellipses have semi-minor axis r, in the lower half-dotted ellipse, and major axis d, across the upper ellipse]
:A = d(πr/2 + h)

:[Bottom right; a 'rhomboid-based prism', or a semi-regular hexagon with identical pairs of vertical, horizontal and diagonal sides, plus three more congruent pairs (one of each dotted) all linking inwards from their own vertex to meet at one of two complimentary points within. The representative horizontal line is marked 'b', a vertical is 'h', a diagonal as 'd'. Between the base horizontal and the lower internal diagonal is a non-'rightangled' angle 'θ']
:A = bh + d(b sinθ + h cosθ)

{{comic discussion}}

[[Category:Math]]

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Talk:2565: Latency

2509: Useful Geometry Formulas