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Surface, Curves and Equations

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If we talk in mathematical terms, the surface Geometry deals with the plane and smooth surfaces. “Carl Friedrich Gauss” gave this concept. In this field the vital role has been played by the Lie groups. Lie groups define many essential elements of the surfaces and also enable us to know about its properties.

A curve is a figure which is not straight like line. Curve geometry may be two-dimensional like Plane Curve and can be three-dimensional like space curve. The meaning of curve basically depends on the context in which it is explained.

Parabola is an example of curve and arc is the segment of curve which is made by two distinct points on the curve.

Those equations which are required to find out the area, volume, surface area of the various mathematical figures are called geometry equations. Some of the formulas are shown below:

Area of Square = m 2; where ‘m’ is its side.

Area of Rectangle = mn; where ‘m’ is length and ‘n’ is breadth.

Area of triangle = (1/2) mn; where ‘m’ is its base and ‘n’ is its height.

Volume of Cube = m 3; where ‘m’ is its sides.

Volume of cylinder = ∏s2 t; where‘s’ is its base and t is its height.

Volume of cone = (1/3) ∏r 2 t; where ‘r’ is its radius and t is its lateral height.

Volume of Sphere = (4/3) (22/7)r 3; where ‘r’ is its radius.

Surface area of cube = 6t 2;where ‘t’ is its side.

Surface area of sphere = 4∏ r 2 ; where ‘r’ is its radius.

Equation of a surface

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In day to day life, we see different types of shapes some of them are round some are Square and many more. Here we will talk about some of the shapes which are present in the Geometry. In the mathematical geometry there are many shapes which are Circle, cone, cylinder, sphere, and pyramid and so on. Here we will see what these shapes actually are and how to find the surface equation of these shapes.

First we will talk about circle. The number of square units which is accurately filled in the interior of a circle is said to be area of circle.

Area = $\pi$r2, where ‘r’ is the Radius of Circle and the value of π is 3.14.

Let’s talk about the cone.

Cone is a three dimensional surface which is having circular base and only one vertex. We can say it has a flat base, and one side is curved. If we want to find the equation of a surface area then we use formula:

Surface area of base = $\pi$ * r2,

Surface area of side = $\pi$ * r * s,

Or Surface area of side = $\pi$ * r * √ (r2 + h2),

This is the surface equation of a cone.

Let’s talk about cylinder. The equation for surface area circular cylinder is calculated by using the following formula:

The total surface area of Right Circular Cylinder = 2$\pi$rh + 2$\pi$r2,

Where, r denotes the radius of a right circular cylinder,

H denotes the height of a right circular cylinder,

Let’s talk about another geometrical shape which is known as pyramid. A Solid object which has its base as Polygon and other sides as triangle is known as Pyramid. Now we will see the formula for finding the LSA and TSA of a pyramid.

Then Lateral surface area of a pyramid is given by:

LSA of a pyramid = μ;

The total surface area of a pyramid = area of the base + 4 * area of triangular face.

This is all about different surface equations of different geometrical figures.

Equation of a curve

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In our day to day life we deal with many types of shapes and figures like lines, circles, ellipses, curve and many more. They all come under the curves in Geometry. A Geometry curve can be defined as a line which is not a Straight Line. There are two types of curve which are as follows:-
1. Open curve: - In open curve, we consider those curves whose close Point do not gather. For example:- parabola, the simple example of curve.
Now, we will see the curve equation. The equation of curve is given by:
⇒y = x2, here y denotes the y – coordinates value and ‘x’ denotes the x – coordinate values.

Figure 1 of parabola
In Parabola example the close points are ‘a’ and ‘b’ and they do not meet at any point.

Figure 2 of hyperbola
In Hyperbola example, the close point’s p, q, r and t are also not meeting at any point.
Now we will talk about close curve.
1. Close curve: - The types of curve whose close points are connected are known as close curve. Close curve does not contain any close points. For example: - circle, ellipse or any other closed curves are the example of close curve.
Let’s talk about the curves in geometry: - As we know that curve can be use in 2 D (plane) curve and 3 D (space). Basically the curves are of two type which are as follows:-
1. Analytic Curve: - In this type of curves generally ellipse, straight line and helices are considered. It contains easy curves so that they can be managed by an equation.
2. Interpolated Curve: - All B spline curves are interpolated curve.
Some curves are inside the curve and are known as interior curves. Some curves are outside the curve and are known as exterior curves. Some are on edges (means on boundary) which are known boundary curves.

Cones

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In Geometry, cone is a three dimensional surface which has circular base and also one vertex. In other word, we can say that it is has flat base, and one side is curved surface of a cone. Cones are not polyhedral because it has a curved surface.
Let’s see some formulas related to the cone. Suppose we want to calculate the surface area of base then the formula is given by:
Surface area of base = ⊼ * r2,
Surface area of side = ⊼ * r * s,
Or Surface area of side = ⊼ * r * √ (r2 + h2),
Volume of a cone = ⊼ * r2 * (h / 3),
Where ‘H’ denotes the height of a cone,
‘R’ denoted the radius of a cone.
‘S’ denoted the side length.
In geometry, cone has a pointy end that is known as vertex or apex of a cone. The flat part of a cone is called as base of a cone. The shape of a cone is conical. If we rotate a triangle then a new shape is formed that is also known as cone. Similarly, if we rotate a triangle along the shortest side then it can be result as right angled triangle.
Now, we will see the different types of properties of a cone.
Properties of a cone are used to define the volume, surface area and the total surface area of a cone. Volume of a cone is given by:
Volume of a cone = r2 * h,
3
Where ‘r’ is the radius the cone’s base and ‘h’ is the height of a cone.
Now surface area of a cone is given by:
SA = ⊼ * r * √ (r2 + h2),
Now total surface area of a cone:
TA = SA + ⊼r2,
Where ‘r’ denotes the radius of cone’s base and SA is the cone’s surface area of face.

Cylinders

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In cone, base of a cone is always perpendicular to its sides. If a line joining the center of Circle of a given cylinder is perpendicular to the base then the cylinder is a Right Circular Cylinder. Let’s see some formulas and properties of cylinder.
If length and radius and length or height is defined for then cylinder then volume of a right circular cylinder is given by:
V = ⊼r2h, where ‘r’ denotes the radius of a cylinder and ‘h’ is the height of cylinder.
The lateral surface area of Cylinders is given by:
A = 2⊼rh (this area is defined for lateral area or without top or bottom),
The total surface area of cylinder with top and bottom is defined as:
A = 2⊼r2+ 2⊼rh,
We can write it as:
A = 2⊼r (r + h),
Properties of a cylinder are given as:
The axis of a right circular cylinder is a line joining the center of cylinder which has a base. Let’s see how these formulas are used?
Let the radius of cylinder be r = 6 inch and h= 4 inch, then we have to find the volume and surface area of cylinder.
As we know that the volume of cylinder = ⊼r2h,
And surface area of cylinder = 2⊼r(r + h),
We know that the value of ⊼ = 3.14,
Given, radius = 6,
Height = 4,
On putting the value we get,
Volume of right circular cylinder = ⊼r2h,
V = 3.14 * (6)2 * 4,
V = 3.14 * 36 * 4,
V = 3.14 * 144,
V = 452.16 inch3,
Surface area of right circular cylinder = 2⊼r (r + h),
SA = 2⊼r (r + h),
= 2 * 3.14 * 6 (6 + 4),
= 2 * 3.14 * 6 (10),
= 2 * 3.14 * 60,
= 376.8 inch2.
This is all about Cylinders.

Plane and straight line

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In mathematical Geometry, a plane is a two dimensional surface and it is used to show two dimensional shapes like Circle, triangle, line etc. The angle lie in between two intersecting plane is a dihedral angle.
A line which is not in the form of a curve is known as Straight Line. As we know, equation of a straight line is given by:
Y = mx + b; or,
Y = mx + c;
Where ‘m’ denotes the Slope (gradient) of a line.
And ‘b’ is the y- intercept.
Let’s see how to find the value of ‘m’ and ‘b’;
To find the value of ‘b’ we have to find where the line intersects the y- axis. Than we have to find the value of ‘m’.
We know that, ‘m’ denotes the Slope of line, and then the value of ‘m’ is:
m = Change in y,
Change in x
Let change in ‘y’ is 6 and change in ‘x’ is 3 then we can find the value of ‘b’ and Slope of a line.
As we know slope of line is:
m = Change in y,
Change in x
So put the value in the given formula:
m = 6
3
Thus, the slope of the line is 6 and the value of b is 3.
So the equation of line by putting the value of slope and b is:
We know that the equation of line is:
Y = mx + b;
Put value of slope and y- intersect we get the equation of line.
Y = 6x + 3;
Let’s see slope of the straight line.
Let the coordinates of a straight line is s1, s2 and t1, t2 and ‘m’ is the slope of a line, and then we use following formula for finding the slope of a straight line.
m = t1 - t2,
s1 - s2
This is all about Plane and straight line.

Loci in Space

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When a common characteristic exists between a Set of coordinates, they are said to define a locus. The locus of points is defined such that all points will have a constant distance from a given straight line, figure, or equality. For example, boundary of Circle can be considered as collection of points that lie at a fixed distance from the Center of Circle. This fixed distance is called as Radius of Circle. To define locus we draw a graph such that passes through all the points that encompass the common characteristic. Locus definition will be correct if the right graphical representations of points are plotted on your Cartesian coordinate system.

First make a plot or graph on the Cartesian plane of the original equation that can represent any curve of mathematics. Suppose we wish to graph the locus of line x = 2. First we graph the given line by drawing a vertically Straight Line passing through x = 2 and parallel to y – axis.

Next we find out the coordinate pairs that define a locus. In our example we have a straight line passing through x = 2 and we are willing to find all those points that are horizontally spaced from a fixed Point you want to find the points that are horizontally spaced 3 units away from the given line. Trace all such points. Locus of all such points is the vertical line, parallel to the given line and can be either of two: x = 5 or x = -1.

Draw these two lines as dotted lines to represent the loci of the points lying on the original equation of the line x = 2.
This is how we plot the locus in space by finding all those points that define it.