Geometry, a branch of Mathematics, deals with various shapes, figures and sizes. It also concerns the properties of these various figures. Ideally, geometry can be diverted into two forms two-dimensional geometry or a plane surface and three-dimensional geometry or solid.
Two-dimensional geometry can be defined as the study of various lines, curves, polygons, flat shapes, etc. These shapes can be drawn on paper and formed using materials. At the same time, a three-dimensional geometry is the study of objects like cylinders, cubes, spheres, etc.
A line or shape that is smoothly bent in a plane or has a turn or bent is defined as a curve. A circle can be called as a curved-shape. For an in-depth explanation of curve meaning, this section explains the definition broadly.
Students need to understand the types of curves to solve the related equations and excel in respective exams.
There are various types of curves coming with an assortment of shapes which are needed in geometrical equations. Some examples of two-dimensional curves are parabolas, circles, hyperbolas, ellipses, sectors, arcs, segments, etc. Shapes like cones, spheres, cylinders can be defined as 3-D curved shapes.
Here are the general types of curves in maths that are important to understand.
This sort of curve changes direction but doesn’t traverse itself. A simple curve can be in an open or closed form. This shape is essential for a defined meaning of curve and its shapes.
A non-simple as the name defines is a form of a curve that crosses its way in a non-systematic way. When it changes direction, the curve also starts to intersect.
Ideally, a curve has two points at the end which doesn't enclose the area inside. This form of the curve is known as an open curve.
When a curve encloses the area inside or region, it has no endpoints. This is also known as a closed curve. This sort of curve is usually formed by connecting two points at the end of an open curve, Shapes like ellipses, circles, etc. can be an excellent example of this shape.
This form of curve ideally points upwards or upward direction. An upward curve can be either convex or concave in shape and direction.
A downward curve points towards downward giving it a curve shape. This curve shape is also known as convex upward curves or concave downward.
A line that is not straight in shape and bent is called a curved line. When the curvature refuses to be zero, then it is defined as curved. This sort of shape is ideally unbroken and even in texture.
After understanding kinds of curves, let’s see the practical use of curves in a mathematical equation.
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In the above figure, one can assume the curve y=f(x) and its ordinates of the x-axis to be x=a and x=b. To understand how to find the area under the curve, students have to assess the area surrounded by the given curve and its ordinates show that x=a and x=b.
Here one can find the definite integral area under the curve is thin like a strip. Taking a random strip of height as Y and width as dx. In the figure above, dA is assumed to be the area.
Now the dA area of the strip is provided with a y dx while a point in the curve that is y is represented through f(x).
The strip areas between curves calculus can also be termed as an elementary area between the x-axis. And the curve is located between x=a and x=b. To find the total is bounded by this curve, one has to consider there is an infinite number of strips. These strips start from x=b to x=a. Adding an elementary area between given strips in the region PQRSP helps find the total area needed.
Applying this formula practically will help in understanding the different types of curves better.
These are basic concepts; students need to refer to quality study materials and reference books for an explanation on types of indifference curves. There are several types of curves which have specific application and formula.
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1. What is the Primary Difference Between Curve and Line?
Ans. Curves are ideally lines having different shapes while a line has a straight shape. A straight line can be shaped into a curve as per need. In Euclidean theory, a straight line has a short distance which is ideally between two points. This idea only exists in Euclidean geometry.
Moreover, a straight line is a short line joining any two points and moving into one direction. But a curve which is bent in shape doesn’t move in a pre-defined direction. They can be bent in a sine wave or circle pattern for fulfilling requirements.
2. What are Plane Algebraic Curves?
Ans. A plane algebraic curve is usually the polynomial with two indeterminates having zero value. Generally, an algebraic curve satisfies a finite set of polynomials in a dimension. The coefficients of the polynomial that is part of field p, then the curve is said to be definite over p. Here P works as an algebraic curve is a finite medium or mark of real numbers.
Considering a composite algebraic curve with complex zero when seen from a topological point forms a surface. This ideology is often termed as a Riemann surface. Algebraic curves defined over other fields have been studied in the ordinary sense. They are often used for cryptography and related services.
3. What are Coordinates in a Curve?
Ans. In two dimensions, coordinates in a point remain stable, and the other coordinate keeps varying. The Cartesian coordinate system, straight lines are usually used as coordinate lines. It is seen that coordinate axes have parallel lines. Moreover, they can be in curves in alternative coordinate axes.
In a polar coordinate belonging to a coordinate curve obtained by investing an ‘r’ constant are the circles are having a centre at derivation. At the same time, a curvilinear coordinate system is referred to as a coordinate curve without lines.