and geometric shapes.  What precisely are these functions in geometric terms?
199917
Take a unit circle and a point p on the circle. Embed the circle to be centered at the origin on a Cartesian xy-plane. The cosine of p, cos(p), is the x-coordinate, or abscissa, of p. The sine of p is the y-coordinate. These functions are the projection functions for the unit circle. \nIt is common to give the points p on the 1-dimensional circle the coordinate referring to the arclength between p and the positive x-axis (as measured clockwise). Thus, the point (1, 0) on the unit circle is given the native coordinate (0), or 0 for short. Inversely, the point 0 in the unit circle has Cartesian coordinates (cos(0), sin(0)). Since the circumference of a circle is 2Pi, a point that is a quarter of the way around the circle from the positive x-axis is given the coordinate 2Pi/4 = Pi/2 and so forth. This system of coordinates is called radians and is the native way of measuring angles, as each angle has a unique radian measure. \nThis relationship can be seen by drawing a radius from the origin to mark the angle and assigning it the radian measure of wherever it intersects the unit circle. There are more analytical reasons as well, as you go on to study calculus.\nThe radius drawn above also gives the relationship between these circular projection functions, and the way one first encounters trigonometric functions, as ratios of the sides of right triangles. Now you should be able to visualize the triangles inside the unit circle (or any larger circle, dependent on the triangle's hypotenuse), and see why the trigonometric functions are cyclic in 2Pi, and derive some properties of the trigonometric functions that are not easily apparent from just looking at right triangles.\nThe tangent function is so named because it is the length of a certain tangent to the circle; a tangent drawn from the positive x-axis to where it meets an extended radius of the circle. You may be able to work out the equation tan(p) = sin(p)/cos(p) from the geometry involved. If you're familiar with the secant function, (1/cos(p)), you may want to know that it is a geometric secant as well. See the sources below for pictures.
506923
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.
490900
Whee, uh, I doubt if this is something you heard about in high school, but here it goes:  A modular form is a complex analytic function on the upper half-plane satisfying a number of conditions, and defined by two complex axes, so that they exist in hyperbolic space. They exhibit infinite symmetry under transformations of the type f(z) -> f[(az+b)/(cz+d)].  Related to elliptic functions, they appear in algebraic topology, string theory, and even in Mandelbrot sets. Here's the website about this last aspect, check out the color pictures:
376353
It's straight ahead geometry.\nDraw a circle with center O.  Draw a diameter horizontally through the circle. Label the left intersect as A and the right intersect as B.  Now draw a vector from O to a point P on the circle.  What tdo you know about triangles AOP and BOP in terms of the angle the vector makes with the diameter?  Use law of sines, law of cosines, trig, etc. to show that angle APO and angle BPO add up to 90 degrees and your done.\n\n\nDoug
59168
cool! I'll think about it\n\n---\n\nhere I am.\nUse the two perpenducular segments as a coordinate system with O on their intersection.\n\nSay: the 4-3 segment defines the x axis, and the 2-6 segment defines the y axis.\nYou know four points belonging to the circle: (-4;0),(0;-2),(3;0) and (0,6)\nThe center of the circle has the mean value between -4 and 3 as its x and the mean value between -2 and 6 as its y (property of chords: the center is on the chord's axis)\n\nso the center's coordinates are C(-1/2;2)\n\nNow you can easily calculate the distance between C and anyone of the four points you know to get the radius.\n\nThen finish finding the diameter\n\n----
431040
In mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. The basic hyperbolic functions are the hyperbolic sine "sinh", and the hyperbolic cosine "cosh", from which are derived the hyperbolic tangent "tanh", etc., in analogy to the derived trigonometric functions. The inverse functions are the inverse hyperbolic sine "arsinh" (also called "arсsinh" or "asinh") and so on.\n\nJust as the points (cos t, sin t) define a circle, the points (cosh t, sinh t) define the right half of the equilateral hyperbola. Hyperbolic functions are also useful because they occur in the solutions of some simple linear differential equations, notably that defining the shape of a hanging cable, the catenary.\n\nThe hyperbolic functions take a real value for real argument called a hyperbolic angle. In complex analysis, they are simply algebraic functions of exponentials, and so are entire.
113395
I'm not familiar with that one.  I know a circle as a "set of points all equidistant from a fixed point in a plane".\n\nI see where you are going with your definition, but I don't think it is worded well.  For one, it doesn't specify that these direction-changing lines are pivoting around a fixed point.  They could just be scattered all over the plane in which case, since they are infinite, you'd end up with a shaded in plane instead of a circle.  For another, they would need to be line segments instead of lines, and the endpoint would be what is creating the closed loop.  And which point on the line segment would they be pivoting around?  If it's one endpoint, then the line seg. would be the radius; if the midpoint, diameter.  Any other point on the line segment would create 2 circles.
642236
geometry:the pure mathematics of points and lines and curves and surfaces\n\nGeometry (from the Greek words Geo = earth and metro = measure) is the branch of mathematics first popularized in ancient Greek culture by Thales (circa 624-547 BC) dealing with spatial relationships
374588
composition means like g(f(x))?\n\ndomain or result is a sub-set of domain of inner function (f),\nrange is a subset of the range of outer function (g)
506476
I'm going to focus on the usual non-euclidean geometry due to Lobachevsky. In that geometry, there are infinitely many lines parallel to a given line and through a given point. Squares (i.e. figures with 4 equal sides and 4 equal angles) don't have right angles. There is a pentagon with all right angles. The area of a triangle can be determined by the sum of the angles in the triangle (it is proportional to the difference of 180 degrees and the sum of the angles). There is a 'natural' unit of distance obtained from angles between parallel lines and perpendicular lines.
502700
There can be several parameterization\n\n1. The specification of a curve, surface, etc., by means of one or more variables which are allowed to take on values in a given specified range. \n\n2. A parameterization of a surface  X(u,v) in uand v is regular if the tangent vectors \ndX/du and dX/dv are always linearly independent. \n[THE DERIVATIVE ARE PARTIAL HERE.]\n\n3. A surface in 3-space can be parameterized by two variables (or coordinates) u and u such that \n\n   (1)  x= x(u,v)\n   (2)  y= y(u,v)\n   (3)  z= z(u,v)\nIf a surface is parameterized as above, then the tangent vectors \n\n (4) Tu =   (dX/du)x^ + (dY/du)x^ + (dZ/du)x^ \n\n (5)  Tv =  (dX/dv)x^ + (dY/dv)x^ + (dZ/dv)x^ \nare useful in computing the surface area and surface integral. \n\n[Tv means T with suffix v, (dX/dv) means partial derivative of X with v and x^ means x with a cap].\n\nSorry, but I can not write euqations so well without equation editor.
442062
Zeroth dimension = point\nFirst dimension = line\nSecond dimension = square\nThird dimension = cube\nFourth dimension = tesseract\nFifth dimension = decateron\n\nThe rests have names (I've lost the website), but what you are listing is the names of polygons (at least after the tesseract), all two-dimensional objects.\n\nA two dimensional object with:\n3 sides = triangle\n4 sides = square\n5 sides = pentagon\n6 sides = hexagon\n7 sides = heptagon\n8 sides = octagon\n9 sides = nonagon\n10 sides = decagon\n\nSee the below for the construction of the rest of the names:
541907
Similarities:\n- They both deal with square roots.\n- They both involve solving for a particular variable.\n- The quadratic equation is used in both of them.\n- They both deal with points and lines (the third-dimensional planes are used mostly in geometry).\n\nDifferences:\n- Geometry doesn't involve negative numbers; algebra does.\n- Geometry for the most part doesn't involve polynomials or factoring; algebra does.\n- Alegbra for the most part doesn't involve shapes; geometry does.\n- Geometry probably uses more formulas. \n\nThe differences were easier to think of. Also, geometry is a lot more visual than algebra because you're looking at shapes most of the time, instead of just numbers.
588357
a circle is meant to be round with same radius\na circle of triangular or some other shape is not possible \nif a circle is changed to tringular shape then it is a triangle but not a circle
638292
This is a good one.\nAccording to the definition in geometry a vertex is a corner.\n\nA triangle has 3 vertices \nA square has 4 vertices  \n \nSo far it is a piece of cake.\n\nSo how many corners does a sphere have? \n\nIf we have to approximate a sphere (actually I’m thinking of a circle but for our argument it is good enough) we would need infinitely many edges (or sides) and therefore would generate infinite number of vertices since a vertex is formed between every two edges. \n\nSo a sphere has infinite number of vertices. \n\nSome people would say that it has a zero vertices since there are no corners to a sphere. But now you know better don’t you?
67785
In "C", it is called the "floor" function. A quick search on wikipedia\n\nhttp://en.wikipedia.org/wiki/Floor_function\n\nshows that mathematicians favour the same terminology.
637407
look std 10 th NCERT mathematics text book... its in the part 2 of  mathematics in geometry section.