Editing 2509: Useful Geometry Formulas
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==Explanation== | ==Explanation== | ||
− | This comic showcases area formulas for | + | {{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 formulas 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. | The illustrations depict the following plane or solid figures, depending on the interpretation. | ||
− | ; Top Left | + | ; Figure Top Left |
− | This illustration is commonly used to depict a three-dimensional sphere, with the ellipse representing a "horizontal" or axial cross-section through the | + | A circle with radius r, with a concentric ellipse drawn over it, sharing its horizontal diameter. The lower half of the ellipse is drawn with a solid line and the upper half with a dotted line. This illustration is commonly used to depict a three-dimensional sphere, with the ellipse representing a "horizontal" or axial cross-section through the centre; the solid lower half of the ellipse represents the "front" of the circumference of this cross-section, while the dotted upper half represents the "back" of the same section, which would be occluded from view if this were a solid shape. |
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The area of the 2D shape on the page is the area of the circle, which is A = πr<sup>2</sup>. This is captioned below the figure. | The area of the 2D shape on the page is the area of the circle, which is A = πr<sup>2</sup>. This is captioned below the figure. | ||
− | + | The 3D sphere commonly depicted by this drawing would have a volume of <sup>4</sup>/<sub>3</sub> πr<sup>3</sup> and a surface area of 4πr<sup>2</sup>. The vertical cross-sectional area of the sphere and the horizontal cross-sectional area would both be the same as the area of the 2D shape on the page, πr<sup>2</sup>. In textbooks a drawing such as this may be captioned with its volume, surface area or the area of the horizontal cross-section depicted by the ellipse; at first glance a reader familiar with such images may assume that the actual caption in the comic is meant to show the formula for the horizontal cross-section. This is not the case for the rest of the figures. | |
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− | The 3D sphere commonly depicted by this drawing would have a volume of <sup>4</sup>/<sub>3</sub> πr<sup>3</sup> and a surface area of 4πr<sup>2</sup>. | ||
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− | + | ; Figure Top Right | |
+ | An isosceles triangle with a vertical axis of symmetry, of height h, with its base replaced by an ellipse with semi-minor axis a (vertical) and semi-major axis b. The lower half of the ellipse forms the outline of the combined shape, while the upper half of the ellipse is depicted with a dotted line. This illustration is commonly used to depict a right circular cone, with the lower half of the ellipse representing the "front" of the bottom surface, and the upper half representing the occluded "back" edge; the drawing may also depict a right elliptic cone. In textbook depictions of a right circular cone, "a" is assumed to be actually equal to "b", but because it represents the dimension "depth" which cannot be drawn on the page, it is instead drawn as shorter than "b" due to parallax. | ||
− | The | + | The area of the 2D shape on the page is the sum of the area of the triangle (bh), and half of the area of the ellipse (<sup>π</sup>/<sub>2</sub> ab), since the upper half of the ellipse overlaps the triangle on the page. The equation for this area is A = 1/2 πab + bh. This is captioned below the figure. |
− | + | The 3D right circular cone commonly depicted by this drawing would have a volume of πr^2*h/3 where r=a=b. The area of the "lower" surface would be πr<sup>2</sup>, while the surface area of the upper conical surface would be πr√(h<sup>2</sup> + r<sup>2</sup>). Neither of these areas correspond with the caption in the comic, nor does the total surface area (the sum of these two). | |
− | + | If we do not assume that a = b, this drawing could also depict a right elliptic cone. The volume of the elliptic cone would be <sup>π</sup>/<sub>3</sub> abh. The area of the lower surface would be πab and the area of the curved upper surface would be <br>2a√(b<sup>2</sup> + h<sup>2</sup>) ∫<sub>0</sub><sup>1</sup> √(<sup>a²h²(t²-1) - b²(a²+h²t²)</sup>/<sub>a²(t²-1)(b²+h²)</sub>) dt. | |
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− | + | <b>Bottom left.</b> 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, <sup>π</sup>/<sub>2</sub> 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 ∫<sub>0</sub><sup>1</sup> √(<sup>1 - t²(1-4r²/d²)</sup>/<sub>1 - t²</sub>) dt. The volume is <sup>π</sup>/<sub>2</sub> rdh. | |
− | 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. | + | <b>Bottom right.</b> 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. | 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== | ==Transcript== | ||
− | :[Four figures in two rows of two, each | + | {{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 | :Useful geometry formulas | ||
− | :[Top left; A circle with | + | :[Top left; A circle with a concentric ellipse sharing its horizontal diameter. The major axis of the ellipse is horizontal. The edge of the ellipse above the major axis is drawn with a dotted line, while the lower edge is drawn with a solid line, similar to textbook depictions of a 3D sphere. The shared semi-major radius is labelled 'r'. ] |
:A = πr² | :A = πr² | ||
− | :[Top right; An ellipse with | + | :[Top right; An ellipse drawn with its major axis horizontal, with two straight lines, one from each end of the major axis, up to a point vertical to the centre of the ellipse, forming an isosceles triangle. The upper edge of the ellipse above the major axis is drawn with a dotted line, while the lower edge is drawn with a solid line, similar to textbook depictions of a right elliptical cone, or mor commonly a right circular cone. The ellipse has its semi-minor axis labelled 'a' and semi-major axis labelled 'b'. The height of the isosceles triangle is labelled 'h'.] |
:A = 1/2 πab + bh | :A = 1/2 πab + bh | ||
− | :[Bottom left; Two ellipses of the same dimensions, with major axes horizontal, drawn vertically one above the other, with vertical lines connecting each end of the major axis of the top ellipse to the corresponding points on the bottom ellipse. The upper edge of the bottom ellipse above the major axis is drawn with a dotted line, while the lower edge is drawn with a solid line, similar to textbook depictions of a right elliptical prism or, more commonly, a right cylinder | + | :[Bottom left; Two ellipses of the same dimensions, with major axes horizontal, drawn vertically one above the other, with vertical lines connecting each end of the major axis of the top ellipse to the corresponding points on the bottom ellipse. The upper edge of the bottom ellipse above the major axis is drawn with a dotted line, while the lower edge is drawn with a solid line, similar to textbook depictions of a right elliptical prism or, more commonly, a right cylinder. The length of the vertical lines is labeled 'h'. The major axis of the upper ellipse is labelled 'd'. The semi-minor axis of the lower ellipse is labelled 'r'.] |
:A = d(πr/2 + h) | :A = d(πr/2 + h) | ||
− | :[Bottom right; Two rectangles of the same vertical and horizontal dimensions, drawn with one offset diagonally to the upper right of the other, with diagonal lines connecting the corresponding vertices, forming a hexagon with opposite sides parallel. The upper right rectangle has its left and bottom sides drawn with dotted lines, and a similar dotted line is used connecting the bottom left corner of the two rectangles, similar to textbook depictions of rhomboid-based right prisms, or more commonly rectangular prisms. | + | :[Bottom right; Two rectangles of the same vertical and horizontal dimensions, drawn with one offset diagonally to the upper right of the other, with diagonal lines connecting the corresponding vertices, forming a hexagon with opposite sides parallel. The upper right rectangle has its left and bottom sides drawn with dotted lines, and a similar dotted line is used connecting the bottom left corner of the two rectangles, similar to textbook depictions of rhomboid-based right prisms, or more commonly rectangular prisms. The lower left rectangle has the length of its lower edge labelled 'b', and the length of the left edge labelled 'h'; the length of the diagonal line connecting the upper left corners of the two rectangles is labelled 'd'. The acute angle between the lower edge of the lower rectangle and dotted diagonal labelled 'θ'] |
:A = bh + d(b sinθ + h cosθ) | :A = bh + d(b sinθ + h cosθ) | ||