what is the geometric theory for spirals?
921616
A spiral is just a monotonically increasing function that defines the radius (r) as a function of the angle (a).  The angle is normally in radians.  The simplest spiral is a linear one:\n\nr = k*a\n\nYou can use any increasing function of r.  \n\nAn exponential spiral: r = e^(k*z) will grow faster for each revolution.\n\nA logrithmic spiral: r = ln(k*a + 1)  grow more slowly with each revolution. \n\nYou can define spirals that are lumpy and maybe not technically spirals such as:\n\nr = a - sin(5a)\n\nWhich makes pentagon shaped loops.
570352
Plain Newtonian Physics and Kepler's Laws; even a circle is a special case of an ellipse with both its focii coinciding (Coordinate Geometry).You may like to know more about properties of ellipses, circles, parabolas, hyperbolas, etc. 3D Coordinate Geometry and spherical trigonometry are pretty helpful.
410236
It's been too long for me.\n\nSample many definitions. Go to MATHWORLD.COM (lower case)\n2 geometric figures are CONGRUENT when one gets transformed into the other by isometry.\n\nStay with it.\n\nGRIZZ
121253
Any on Analytic Geometry. My suggestion is "Analytic Geometry" by Charles Lehmann or "Modern Analytic Geometry" by William Wooton.
2767
The Pythagorean Theorem definitely: a^2+b^2=c^2 \n\nPoint slope form: y - y1 = m(x-x1) \nStandard y equation: y=mx+b\n...Sorry, cant think of any more. \nSee, the rest of the equations that i could give you are not the ones EVERYONE should know :)
822597
in the plane,\nthere are two kinds of curvature,\nthe regular one, which is the norm of the second derivative, if the curve has unit tangent.\nand the signed curvature, which in addition measures if the first derivative and the second derivative preserve orientation.\ns
619341
I hope I get this right, it has been a very long time since I took geometry and I was not very good at it. But here goes. Geometry is the mathematical study of lines, angles, circles, circumferences,triangles and the formulas needed to deduce these ordinary objects correct degree of angle or the volume contained within the circumference of an object. I think I may have left something out, but right now that is all I remember.
274232
Euclidean geometry is the study of geometry as defined by the axioms of Euclid. You can find those axioms here: http://en.wikipedia.org/wiki/Euclidean_Geometry .\nIt was noted by Klein that Euclidean geometry was actually just the study of one particular group of symmetries. His Erlangen program of 1872 revolutionized geometry by encouraging the stuidy of non-Euclidean geometries by simply studying the geometries produced by any arbitrary group.\nGauss and Riemann realized that while Euclidean geometry did not deal with a fundamental unit of length, the study of geometry on curved surfaces did introduce fundamental units of length (this was later applied in general relativity). The geometry on these surfaces, such as the sphere where parallel lines can meet at either 2 points or none, definitely did not satisfy Euclid's axioms, so they were also non-Euclidean. The Riemann sphere's relation to the complex plane made this geometry very useful. The projective plane is a non-Euclidean geometry that has found many applications as well. \nBasically, non-Euclidean geometry pops up whenever a surface (or an arbitrary smooth manifold) has non-zero Gaussian curvature. Things can get pretty wild in more than 2 dimensions. You can read about more non-Euclidean geometries oand their applications on this page: http://en.wikipedia.org/wiki/Non-euclidean_geometries
801553
lol.\nfind formula for area of spiral.  subtract smaller from bigger.  sry no more help.
228730
You can describe rotations and the effect they have on each point in the figure mathematically by means of simple trigonometric equations which can be condensed into a matrix. Imagine (or better, draw) a point on the plane in coordinate system X0,Y0, located at X0=1, Y0=0. Draw a line to the point from the origin. Now draw another coordinate system X1,Y1 with the same origin as X0,Y0 but rotated CW by an angle Theta. (Make Theta small, say 30 degrees.) What are the coordinates of our point in the new system? The answer is a function of Theta. We find that the point is above the X1 axis and nearer the Y1 axis. Specifically, X1 = X0*cosine(Theta) and Y1 = X0*sine(Theta). \nNow if you repeat this with a new point at X0=0, Y0=1, you find that X1 = -Y0*sine(Theta) and Y1 = Y0*cosine(Theta).\nNow the good part. You can put these four equations together to find X1,Y1 for any value of X0,Y0 (not just points located on one axis or the other). The result is\nX1=X0*cosine(Theta) - Y0*sine(Theta)\nY1=X0*sine(Theta) + Y0*cosine(Theta)\nNext you form a rotation matrix R from the Theta-dependent parts of these equations:\nR=[cosine(Theta)___ -sine(Theta)]\n__[sine(Theta) ___ cosine(Theta)] \nNow when you multiply any vector P0, representing any point in X0,Y0 coordinates, by this matrix R using the rules of matrix multiplication, you are actually using the above two equations, and the result is a vector P1 which is the same point represented in X1,Y1 coordinates. Expressed compactly, P1 = R*P0. P0 is said to be transformed by R to P1.\nOf course, this is a 2-dimensional transformation, and R is a 2-by-2 matrix. Sophisticated 3-D computer graphics use a 3-by-3 matrix implementing very similar equations to do 3-D transformations. This is how you can see convincing 3-D models of objects in rotation.
1009048
Major classifications are spiral, elliptical, and irregular.  Some also include lenticular (lens shaped).\n\nSpiral galaxies are similar to the Milky Way or Andromeda -- an elliptical or spherical central bulge with long, winding spiral arms swirling around the nucleus.  There are some sub-classes within spirals, like barred spirals, too.\n\nElliptical galaxies are just that -- huge egg-shaped clusters of stars.  Lenticular (lens-shaped) could be considered a middle form between a spiral and an elliptical.\n\nIrregular galaxies take in everything else.
1136799
The general formula for an exponential or logartithmic spiral is given by:\n\nr = a*e^(b*theta)\n\nFor your case, 'r' is the same as 'y', and (a = 1) & (b = -1)\n\nr=1 implies:\n\n1 = e^(-theta)\n\nthis is only possible when theta = 0.\n\nThe arc length of your exponential spiral (from r = some value, to the origin) is given by:\n\ns = (a/b)*sqrt(1+b^2)*e^(b*theta)\n\nHence, for your problem:\ns = (1/-1)*sqrt(1+(-1^2))*e^(-0)\ns = -1*sqrt(2)\ns = -1.414\n\nHence, according to my calculations, the arc length or distance is equal to sqrt(2), or roughly 1.414 units.\n\nP.S. I think the negative sign indicates that it is a clockwise spiral
729757
hyperbolic plane, you can check what the different models for hyperbolic geometry are.\n\nanotherone, would be Riemannian geometry, which is the basis for relativity theory.\n`
932841
This might help --> http://www.uwm.edu/~ericskey/TANOTES/Geometry/node15.html
1165642
One way to reason is this.  Let n be any even integer greater than or equal to 4.  Divide the circle into n equal wedges (by drawing n/2 diameter lines with angles between successive lines equal to 360/n degrees).  Now, cut the wedges out and make a "rectangle" with them: n/2 wedges with the points downward and the arcs on top of adjacent pairs of wedges touching each other, then the other n/2 wedges with the points upward and adjacent arcs touching, filling the spaces of the first n/2 wedges.  What you end up with is almost a rectangle with top and bottom each made of n/2 360/n-degree arcs, and the lateral sides slanting slightly, made from the long edges of the wedges on each end.\n\nYou can estimate the area of this figure (which is the same as the area of the original circle of radius r) by pretending it IS a rectangle.  The length is the circumference of the circle, or 2*pi*r/2 (since half the circumference is the top, half is the bottom) (by definition of pi).  The width is the length of a wedge, or r.  So the area is about pi*r^2.  \n\nTo turn the estimate into an exact value, consider what happens as n goes to infinity, that is, it increases without bound.\n\nAs n goes to infinity, the slanted edges aproach being verticle and the number of arcs making up the top and bottom approaches infinity, and the arcs themselves approach straight horizontal lines.  Thus, the figure approaches a rectangle of exactly area pi*r^2.  However, for each n, you have the same circle of the same area, so the area of that circle is pi*r^2.
1045681
If your curve is the upper half of an ellipse, then when you rotate it, you have half an ellipsoid.\n\nWikipedia has the formula for the surface are of an ellipsoid, which (surprize!) involve eliptic integrals.
13252
Generally, non-euclidean geometries are geometries that use a different fourth postulate, that, in Euclidean geometry, says that corresponding angles of a transversed parrallel lines are equal.  This basically means that parrallel lines never meet.  There are two other possible postulates.  One is that parrallel lines do meet, i.e. that space is like a globe where longitudinal lines meet at the poles.  The other is that other lines, besides parallel lines don't meet. In this situation space is shaped like a saddle, and lines get farther and farther apart.  Those are the two I know.  There are three other Euclidean postulates (e.g. you can draw exactly one circle through a point in the center and one on the rim) that someone might be able to change, but I am not aware that anyone has done that.