question: How can you have three 90 degree angles all joining up on the globe or other round object? And why can’t you do it on a flat surface?
comment: The lines aren't straight. They curve with the sphere. You can do something similar on a flat surface by making a "triangle" with bowed out edges such that every point is a right angle. If you adhere to keeping all edges straight in three dimensional space, you would run into the same problems you would on a flat surface in that they would never again intersect.
passages: where is the focal length, is the depth of the dish (measured along the axis of symmetry from the vertex to the plane of the rim), and is the radius of the rim. They must all be in the same unit of length. If two of these three lengths are known, this equation can be used to calculate the third.
   BULLET::::- Each point is associated with a unique line, called the "polar line" of the point, which is the line on the plane through the centre of the sphere and perpendicular to the diameter of the sphere through the given point.
  
  As there are two arcs ("line segments") determined by a pair of points, which are not antipodal, on the line they determine, three non-collinear points do not determine a unique triangle. However, if we only consider triangles whose sides are minor arcs of great circles, we have the following properties:
   When defining a form, an important factor is the continuity between surfaces - how smoothly they connect to one another.
  
  One example of where surfacing excels is automotive body panels. Just blending two curved areas of the panel with different radii of curvature together, maintaining tangential continuity (meaning that the blended surface doesn't change direction suddenly, but smoothly) won't be enough. They need to have a continuous rate of curvature change between the two sections, or else their reflections will appear disconnected.
  
  The continuity is defined using the terms
  
  BULLET::::- G0 – position (touching)
  
  BULLET::::- G1 – tangent (angle)
   In the Euclidean plane, angles about a point add up to 360°, while interior angles in a triangle add up to 180° (equivalently, "exterior" angles add up to 360°). However, on a convex polyhedron the angles at a vertex on average add up to less than 360°, on a spherical triangle the interior angles always add up to more than 180° (the exterior angles add up to "less" than 360°), and the angles in a hyperbolic triangle always add up to less than 180° (the exterior angles add up to "more" than 360°).
   In spherical geometry, the sum of angles inside a triangle is greater than 180° (in this example 270°, having arrived back at the north pole a 90° angle to the departure path) unlike on a flat surface, where it is always exactly 180°.
  
  Section::::Effects and empirical evidence.:Grids are distorted by spherical ground.
   If both the side lengths ( and ) of the base of the pyramid and the distance () from the center of the base rectangle to the apex of the pyramid (the center of the sphere) are known, then the above equation can be manipulated to give
  
  The solid angle of a right -gonal pyramid, where the pyramid base is a regular -sided polygon of circumradius , with a 
  
  pyramid height is
  
  The solid angle of an arbitrary pyramid with an -sided base defined by the sequence of unit vectors representing edges can be efficiently computed by:
   Section::::Other methods.:With a marked ruler.
  
  Another means to trisect an arbitrary angle by a "small" step outside the Greek framework is via a ruler with two marks a set distance apart. The next construction is originally due to Archimedes, called a "Neusis construction", i.e., that uses tools other than an "un-marked" straightedge. The diagrams we use show this construction for an acute angle, but it indeed works for any angle up to 180 degrees.
  
  This requires three facts from geometry (at right):
  
  BULLET::::1. Any full set of angles on a straight line add to 180°,
   Imagine for a moment what will happen to the lines of latitude and longitude when they are projected from the sphere onto the flat plane. The lines of latitude are all parallel to the equator, so they will become perfect circles centered on the origin . And the lines of longitude will become straight lines passing through the origin (and also through the "point at infinity", since they pass through both the north and south poles on the sphere).
   where "f" is the fraction of the sphere's area which is enclosed by the triangle. For example, suppose that we draw a triangle on the Earth's surface with vertices at the North Pole, at a point on the equator at 0° longitude, and a point on the equator at 90° West longitude. The great circle line between the latter two points is the equator, and the great circle line between either of those points and the North Pole is a line of longitude; so there are right angles at the two points on the equator. Moreover, the angle at the North Pole is also 90° because the other two vertices differ by 90° of longitude. So the sum of the angles in this triangle is . The triangle encloses 1/4 of the northern hemisphere (90°/360° as viewed from the North Pole) and therefore 1/8 of the Earth's surface, so in the formula ; thus the formula correctly gives the sum of the triangle's angles as 270°.
   Other open surfaces and surfaces closed in one direction, and all surfaces closed in both directions, can't be flattened without deformation. A hemisphere or sphere, for example, can't. Such surfaces are curved in both directions. This is why maps of the Earth are distorted. The larger the area the map represents, the greater the distortion. Sheet metal surfaces which lack a flat pattern must be manufactured by stamping using 3D dies (sometimes requiring multiple dies with different draw depths and/or draw directions), which tend to be more expensive.
  
  Section::::Surface patches.
   For example, the billiard in a square is related in this way to the billiard on the flat torus constructed from four copies of the square; the billiard in an equilateral triangle gives rise to the flat torus constructed from an hexagon. The billiard in a "L" shape constructed from squares is related to the geodesic flow on a square-tiled surface; the billiard in the triangle with angles formula_177 is related to the Veech surface constructed from two regular pentagons constructed above.
  
  Section::::Geodesic flow on translation surfaces.:Relation with interval exchange transformations.
   The solid angle of the complement of the cone (picture a melon with the cone cut out) is clearly:
  
  This is also the solid angle of the part of the celestial sphere that a Terran astronomical observer positioned at latitude can see as the earth rotates. At the equator you see all of the celestial sphere, at either pole only one half.
  
  The solid angle subtended by a segment of a spherical cap cut by a plane at angle from the cone's axis and passing through the cone's apex can be calculated by the formula:
   For convex two-dimensional shapes, the centroid can be found by balancing the shape on a smaller shape, such as the top of a narrow cylinder. The centroid occurs somewhere within the range of contact between the two shapes (and exactly at the point where the shape would balance on a pin). In principle, progressively narrower cylinders can be used to find the centroid to arbitrary precision. In practice air currents make this infeasible. However, by marking the overlap range from multiple balances, one can achieve a considerable level of accuracy.
  
  Section::::Locating.:Of a finite set of points.
   Every conic surface is ruled and developable. In general, a conical surface consists of two congruent unbounded halves joined by the apex. Each half is called a nappe, and is the union of all the rays that start at the apex and pass through a point of some fixed space curve. (In some cases, however, the two nappes may intersect, or even coincide with the full surface.) Sometimes the term "conical surface" is used to mean just one nappe.
   The point of intersection for all four classes on this plot occurs when the object's axes "a":"b":"c" have ratios of "R":"R":1, or 9:6:4 when "R"=. Make axis "b" any shorter and the object becomes "prolate". Make axis "b" any longer and it becomes "oblate". Bring "a" and "c" closer to "b" and the object becomes "equidimensional". Separate "a" and "c" further from "b" and it becomes "bladed". 
  
  For example, the convex envelope for some humans might plot near the black dot in the upper left of the figure.
  
  Section::::See also.
  
  BULLET::::- Aspect ratio between long and short
   Because of the powers of two in the equation there are two possible solutions for the normal direction. Thus some prior information of the geometry is needed to define the correct normal direction. The normals are directly related to angles of inclination of the line on the object surface. Thus the normals allow the calculation of the relative surface heights of the line on the object using a line integral, if we assume a continuous surface.
   From the above angle sum formula we can also see that the Earth's surface is locally flat: If we draw an arbitrarily small triangle in the neighborhood of one point on the Earth's surface, the fraction "f" of the Earth's surface which is enclosed by the triangle will be arbitrarily close to zero. In this case the angle sum formula simplifies to 180°, which we know is what Euclidean geometry tells us for triangles on a flat surface.
  
  Section::::Triangles in construction.
   It is possible to trisect an arbitrary angle by using tools other than straightedge and compass. For example, neusis construction, also known to ancient Greeks, involves simultaneous sliding and rotation of a marked straightedge, which cannot be achieved with the original tools. Other techniques were developed by mathematicians over the centuries.
  
  Because it is defined in simple terms, but complex to prove unsolvable, the problem of angle trisection is a frequent subject of pseudomathematical attempts at solution by naive enthusiasts. These "solutions" often involve mistaken interpretations of the rules, or are simply incorrect.
  
  Section::::Background and problem statement.
   Section::::Special cases.
  
  The sphere is the only surface where both sheets of the focal surface degenerate to a single point.
  
  Both sheets of the focal surface of a Dupin cyclide degenerate to curves. These curves form a pair of anticonics, i.e., an ellipse and a hyperbola in perpendicular planes, each passing through the respective foci of the other curve. For the torus, a special Dupin cyclide, the focal ellipse becomes a circle and the focal hyperbola degenerates to the axis of the circle.
   In an acute triangle the circumcenter divides the interior perpendicular bisectors of the two shortest sides in equal proportions. In an obtuse triangle the two shortest sides' perpendicular bisectors (extended beyond their opposite triangle sides to the circumcenter) are divided by their respective intersecting triangle sides in equal proportions.
  
  For any triangle the interior perpendicular bisectors are given by formula_17 formula_18 and formula_19 where the sides are formula_20 and the area is formula_21
  
  Section::::Bisectors of the sides of a polygon.:Quadrilateral.
   Trilateration in general is calculating with triangles of known distances/sizes, mathematically a very sound system. In a triangle, the angles can be derived if one knows the length of all sides, (see congruence), but the length of the sides cannot be derived based on all of the angles, not without knowing the length of at least one of the sides (a baseline) (see similarity).
  
  In 3D, when four or more angles are in play, locations can be calculated from "n" + 1 = 4 measured angles plus one known baseline or from just "n" + 1 = 4 measured sides.
   However, the equator of a Bricard octahedron is not required to lie in a plane; instead, it can be a skew quadrilateral. Even for Bricard octahedra constructed to have a flat equator, the equator generally does not remain flat as the octahedron flexes. However, for some Bricard octahedra, such as the octahedron with an antiparallelogram equator shown in the illustration, the symmetries of the polyhedron cause its equator to remain planar at all times.
  
  Section::::Additional properties.
   Let OABC be the vertices of a tetrahedron with an origin at O subtended by the triangular face ABC where formula_17 are the vector positions of the vertices A, B and C. Define the vertex angle to be the angle BOC and define , correspondingly. Let formula_18 be the dihedral angle between the planes that contain the tetrahedral faces OAC and OBC and define formula_19, formula_20 correspondingly. The solid angle subtended by the triangular surface ABC is given by
   Section::::Surface tension and shape of a water drop.
  
  Perhaps the most familiar physical manifestation of the 3-dimensional isoperimetric inequality is the shape of a drop of water. Namely, a drop will typically assume a symmetric round shape. Since the amount of water in a drop is fixed, surface tension forces the drop into a shape which minimizes the surface area of the drop, namely a round sphere. Thus the round shape of the drop is a consequence of the phenomenon of surface tension. Mathematically, this phenomenon is expressed by the isoperimetric inequality.
  
  Section::::Isoperimetric inequality in the plane.
   Section::::Registration.
  
  One common problem encountered in geometry processing is how to merge multiple views of a single object captured from different angles or positions. This problem is known as registration. In registration, we wish to find an optimal rigid transformation that will align surface formula_22 with surface formula_23. More formally, if formula_24 is the projection of a point "x" from surface formula_22 onto surface formula_23, we want to find the optimal rotation matrix formula_27 and translation vector formula_28 that minimize the following objective function:
  
  formula_29
  
raw_presuppositions: Three 90 degree angles that join up on a round object are straight lines.
raw_corrections: Three 90 degree angles that join up on a round object aren't straight lines.