### "there is geometry in humming of the string and there is music in the spacing of sphere"

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### coordinate geometry for dummies

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### Coordinate Geometry

Do you recall what a plane is? A plane is any flat surface which can go on infinitely in both directions. Now, if there is a point on a plane, you can easily locate that point with the help of coordinate geometry. Using the two numbers of the coordinate geometry, a position of any point on the plane can be found. Let us know more!
Coordniate geometry is also known as analytic geometry. A coordinate geometry is a branch of geometry where the position of the points on the plane is defined with the help of an ordered pair of real numbers also known as coordinates. In 3D coordinate geometry three dimensional objects are analyzed. When we want to find volume of a cylinder or sphere 3d gerometry is needed.

Another set of formulas related to triangle and slopes are :

Equations of parabola, hyperbola, ellipse and circle can be defined using co-odinate geometry. They all have a corresponding figure and equations :

The general equation of all second degree curves is :

When f(n) is the value of nth term in the arithmetical sequence.

## Circle

Circle is the locus of the points equidistance from a given point which can be called the centre of the circle. The definition of locus can be stated as :
The curve or fugure that is the formed by all the points satisfying some mathematical conditions of the relations between coordinates and points. This curve may be a straight line, an asymtote or other complicated curves.

### 3D geometry

The equation of plane in three geometry is the extension of straight line in 2D geometry.

### Algebraic geometry

Algebraic geometry is a sub-branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros.

## Conic section

We can generate all kinds of curves like circles, parabola and hyperbola from bisecting a cone in various direction.

There is a difference between parabola and ecclipse. A parabola has one focus whereas ecclipse and hyperbola has two foci.

## Escher's reprentation of hyperbolic geometry

Escher has used a particular representation of hyperbolic geometry in which the entire ‘universe’ of the hyperbolic plane is ‘squashed’ into the interior of a circle in an ordinary Euclidean plane. The bounding circle represents ‘infinity’ for this hyperbolic universe We can see that, in Escher’s picture, the Wsh appear to get very crowded as they get close to this bounding circle. But we must think of this as an illusion. Imagine that you happened to be one of the fish. Then whether you are situated close to the rim of Escher’s picture or close to its centre, the entire (hyperbolic) universe will look the same to you. The notion of ‘distance’ in this geometry does not agree with that of the Euclidean plane in terms of which it has been represented. As we look down upon Escher’s picture from our Euclidean perspective, the Wsh near the bounding circle appear to us to be getting very tiny. But from the ‘hyperbolic’ perspective of the white or the black fish themselves, they think that they are exactly the same size and shape as those near the centre. Moreover, although from our outside Euclidean perspective they appear to get closer and closer to the bounding circle itself, from their own hyperbolic perspective that boundary always remains infinitely far away. Neither the bounding circle nor any of the ‘Euclidean’ space outside it has any existence for them. Their entire universe consists of what to us seems to lie strictly within the circle

In more mathematical terms, how is this picture of hyperbolic geometry constructed? Think of any circle in a Euclidean plane. The set of points lying in the interior of this circle is to represent the set of points in the entire hyperbolic plane. Straight lines, according to the hyperbolic geometry are to be represented as segments of Euclidean circles which meet the bounding circle orthogonally—which means at right angles. Now, it turns out that the hyperbolic notion of an angle between any two curves, at their point of intersection, is precisely the same as the Euclidean measure of angle between the two curves at the intersection point. A representation of this nature is called conformal. For this reason, the particular representation of hyperbolic geometry that Escher used is sometimes referred to as the conformal model of the hyperbolic plane. (It is also frequently referred to as the Poincare´ disc.

The same Escher picture as previous figure but with hyperbolic straight lines (Euclidean circles or lines meeting the bounding circle orthogonally) and a hyperbolic triangle, is illustrated. Hyperbolic angles agree with the Euclidean ones. The parallel postulate is evidently violated and the angles of a triangle sum to less than π
there is actually something particularly elegant and remarkable about what does happen when we add up the angles of a hyperbolic triangle: the shortfall is always proportional to the area of the triangle. More explicitly, if the three angles of the triangle are a, b, and g, then we have the formula (found by Johann Heinrich Lambert 1728–1777)
π - (α + β + γ ) = C ∇ where ∇ is the area of the triangle and C is some constant.
In fact, I have not quite Wnished my description of hyperbolic geometry in terms of this conformal representation, since I have not yet described how the hyperbolic distance between two points is to be deWned (and it would be appropriate to know what ‘distance’ is before we can really talk about areas). Let me give you an expression for the hyperbolic distance between two points A and B inside the circle. This is
log [ QA.PB/QB.PA] where
where P and Q are the points where the Euclidean circle (i.e. hyperbolic straight line) through A and B orthogonal to the bounding circle meets this bounding circle and where ‘QA’, etc., refer to Euclidean distances (see Fig below). If you want to include the C of Lambert’s area formula (with C = 1), just multiply the above distance expression by C != 2 (the reciprocal of the square root of C)] For reasons that I hope may become clearer later, I shall refer to the quantity C^(-1/2) as the pseudo-radius of the geometry.

In the formula natural logarithm ( base e) has been used. Hyperbolic geometry, with this deWnition of distance, turns out to have all the properties of Euclidean geometry apart from those which need the parallel postulate. We can construct triangles and other plane Wgures of diVerent shapes and sizes, and we can move them around ‘rigidly’ (keeping their hyperbolic shapes and sizes from changing) with as much freedom as we can in Euclidean geometry, so that a natural notion of when two shapes are ‘congruent’ arises, just as in Euclidean geometry, where ‘congruent’ means ‘can be moved around rigidly until they come into coincidence’. All the white Wsh in Escher’s woodcut are indeed congruent to each other, according to this hyperbolic geometry, and so also are all the black Wsh.

## Projective model of hyperbolic plane

There are indeed other representations of hyperbolic geometry in terms of Euclidean geometry, which are distinct from the conformal one that Escher employed. One of these is that known as the projective model. Here, the entire hyperbolic plane is again depicted as the interior of a circle in a Euclidean plane, but the hyperbolic straight lines are now represented as straight Euclidean lines (rather than as circular arcs). There is, however, a price to pay for this apparent simplification, because the hyperbolic angles are now not the same as the Euclidean angles, and many people would regard this price as too high. For those readers who are interested, the hyperbolic distance between two points A and B in this representation is given by the expression 1/2 {log[RA.SB/RB.SA]}

(taking C = 1, this being almost the same as the expression we had before, for the conformal representation), where R and S are the intersections of the extended straight line AB with the bounding circle. This representation of hyperbolic geometry, can be obtained from the conformal one by means of an expansion radially out from the centre by an amount given by

where R is the radius of the bounding circle and r(c) is the Euclidean distance out from the centre of the bounding circle of a point in the conformal representation . Escher's picture described before has been transformed from the conformal to the projective model using this formula. (Despite lost detail, Eseher's precise artistry is still evident.) Though less appealing this way, it presents a novel viewpoint!

All the geometry models can be reduced to few axioms and notions. Projective geometry is the geometry which properties are derived by continuous projective transformations. These properties remain invariant under these projective transformations. The above model of projective hyperbolic model is this particular kind of projective geometry. Let us explore more about projective geometry:
A projective space is a kind of general vector space. The vector space itself is ‘almost’ a bundle over the projective space. If we remove the origin of the vector space, then we do get a bundle over the projective space, the Wbre being a line with the origin removed; alternatively, as with the particular example of B(c) , we can 'blow up' the origin of the vector space. (I shall come back to this in a moment.) Projective spaces have a considerable importance in mathematics and have a particular role to play in the geometry of quantum mechanics and also in twistor theory. It is appropriate, therefore, that I comment on these spaces briefly here.
The idea of a projective space appears to have come originally from the study of perspective in drawing and painting, this being taken within the context of Euclidean geometry. Recall that, in the Euclidean plane, two distinct lines always intersect unless they are parallel. However, if we draw a picture, on a vertical piece of paper, of a pair of parallel lines receding into the distance on a horizontal plane (say of the boundaries of a straight road), then we Wnd that in the drawing, the lines appear to intersect at a ‘vanishing point’ on the horizon (see Figure below). Projective geometry takes these vanishing points seriously, by adjoining 'points at infinity' to the Euclidean plane which enable parallel lines to intersect at these additional points.
There are many theorems about lines in ordinary Euclidean 3-space which are awkward to state because of exceptions having to be made for parallel lines. In second Figure, I depict two remarkable examples, namely the theorems of Pappos 12 (found in the late 3rd century AD) and of Desargues (found in 1636). In each case, the theorem (which I am stating in ‘converse’ form) asserts that if all the straight lines indicated in the diagram (9 lines for Pappos and 10 for Desargues) intersect in triples at all but one of the points marked with black spots (there being 9 black spots in all for Pappos and 10 in all for Desargues), then the triple of lines indicated as intersecting at the remaining black spot do in fact have a point in common. However, stated in this way, these theorems are true only if we consider

that a triple of mutually parallel lines are counted as having a point in common, namely a 'point at infinity'. With this interpretation, the theorems remain true when the lines are parallel. They also remain true even if one of the lines lies entirely at infinity. Thus, the theorems of Pappos and Desargues are more properly theorems in projective geometry than in Euclidean geometry.

How do we construct an n-dimensional projective space P(n)? The most immediate way is to take an (n + 1)-dimensional vector space V(n+1), and regard our space P(n) as the space of the 1-dimensional vector subspaces of V(n+1). (These 1-dimensional vector subspaces are the lines through the origin of V(n+1).) A straight line in P(n) (which is itself an example of a P1) is given by a 2-dimensional subspace of V(n+1) (a plane through the origin), the collinear points of Pn arising as lines lying in such a plane (Figure below). There are also higher-dimensional flat subspaces of Pn, these being projective spaces P(r) contained in P(n) (r < n). Each P(r) corresponds to an (r+1)-dimensional vector subspace of Vn+1.
How do we construct an n-dimensional projective space Pn? The most immediate way is to take an (n + 1)-dimensional vector space Vnþ1, and regard our space Pn as the space of the 1-dimensional vector subspaces of Vn+1. (These 1-dimensional vector subspaces are the lines through the origin of Vn+1.) A straight line in Pn (which is itself an example of a P1) is given by a 2-dimensional subspace of Vn+1 (a plane through the origin), the collinear points of Pn arising as lines lying in such a plane (Figure below). There are also higher-dimensional flat subspaces of Pn, these being projective spaces Pr contained in Pn (r < n). Each Pr corresponds to an (r + 1)-dimensional vector subspace of V(n+1).

To construct n-dimensional projective space Pn, take an (n + 1)- dimensional vector space Vn+1, and regard Pn as the space of the 1-dimensional vector subspaces of Vn+1 (lines through the origin of Vn+1). A straight line in Pn is given by a 2-dimensional subspace of Vn+1 (plane through origin), collinear points of Pn arising as lines through O in such a plane. This applies both to the real case (RPn) and the complex case (CPn). The geometry of RP2 formalizes the procedures of perspective in pictorial representation: consider the artist’s eye to be at the origin O of V3, taking V3 as the artist’s ambient Euclidean 3-space. A light ray through O is viewed by the artist as single point. What the artist depicts as a ‘straight line’ (RP1 in RP2) (on any particular choice of artist's canvas) indeed corresponds to the plane (V2) joining that line to O. Pairs of planes through O always intersect, even when joining parallel lines in V3 to O. (For example, the two bottom boundary lines in the left-hand picture play the role of the road boundaries of Figure above.)
Imagine that the artist paints an accurate picture of the perceived scene on some canvas that coincides with some particular flat plane (not through O). Any such plane will capture only part of the entire P2. It will certainly not intersect those light rays that are parallel to it. But several such planes will provide an adequate ‘patchwork’ covering the whole of P2 (three will suffice). Parallel lines in one such plane, will be depicted as lines with a common vanishing point in another. We can consider either real projective spaces, Pn = RPn, or complex ones, Pn = CPn. We have already considered one example of a complex projective space, namely the Riemann sphere, which is CP1. Recall that the Riemann sphere arises as the space of ratios of pairs of complex numbers (w, z), not both zero, which is the space of complex lines through the origin in C2. More generally, any projective space can be assigned what are called homogeneous coordinates. These are the coordinates z^0, z^1, z^2 , . . . , z^n for the (n + 1)-dimensional vector space Vn+1 from which Pn arises, but the 'homogeneous coordinates' for Pn are the n independent ratios
z^0:z^1:z^2:z^3:......z^n
(where the zs are not all zero), rather than the values of the individul zs themselves. If the z^r are all real, then these coordinates describe RPn, and the space Vn+1 can be identiWed with Rn+1 (space of n+1 real numbers;). If they are all complex, then they describe CPn, and the space Vnþ1 can be identined with Cn+1 (space of n + 1 complex numbers; )
Since we exclude the point O = (0, 0, . . . , 0) from the allowable homogeneous coordinates, the origin of Rn+1 or Cn+1 is omitted (to give Rnþ1 - O or Cn+1 - O) when we think of it as a bundle over, respectively, RPn or CPn. The fibre, therefore, must also have its origin removed. In the real case, this splits the Wbre into two pieces (but this does not mean that the bundle splits into two pieces; in fact, Rn+1 - O is connected, when n > 0). In the complex case, the fibre is C - O (often written C*), which is connected. In either case, we may prefer to reinstate the origin in the Wbre, so that we get a vector bundle. But if we do this, then this amounts to more than simply putting the origin back into Rn+1 or Cn+1. As with the particular case of C2, considered above, we must put back the origin in each fibre separately, so that the origin is ‘blown up’. The bundle space becomes Rn+1 with an RPn inserted in place of O, or Cn+1 with a CPn in place of O.
In the complex case, we can also consider the unit (2n + 1)-sphere S2nþ1 in Cn+1, just as we did in the particular case n = 1 when constructing the Clifford bundle. Each Wbre intersects S2n+1 in a circle S1, so now we obtain S2n+1 as an S1 bundle over CPn. This structure underlies the geometry of quantum mechanics—although this beautiful geometrical fact impinges only infrequently on the thinking of quantum physicists—where we shall Wnd that the space of physically distinct quantum states, for an (n+1)-state system, is a CPn. In addition, there is a quantity known as the phase, which is normally thought of as being a complex number of unit modulus (e^(iθ), with y real; ), whereas it is really a twisted unit-modulus complex number.15 These matters will be returned to at the end of this chapter, and when we consider quantum mechanics.
the symbol here is the meaning of exponentiation.

## Thalse theorem

Thalse theorem is a theorem about dissecting a triangle. The theorem goes as follows:

## Arc and angle

There is a relationship between arc length and angle of a cirle. A total of 2πr units of length span 2π radians of the circle. So if an arc of length s makes an angle θ then it will span [2πs/2πr] = θ radians. so we get
S = rθ

Area of a triangle can be calculated using the coordinates of the vertices :

Some useful terms and figures :

The distance of a point P(x0, y0) from a line ax+by+c = 0 is computed using

### Reference materials:

Classical electromagnetic theory
Analysis of matter by Russell
Quantum mechanics
Grand Design by Stephen Hawking
Higher Engineering Mathematics ( PDFDrive.com ).pdf
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