Parabola

In mathematics, a parabola is a special kind of plane curve shaped like a U. It is perfectly mirrored on both sides.

One way to describe a parabola is by using a point called the focus and a line called the directrix. The parabola is the set of all points that are the same distance from this focus point and the directrix line.

Parabolas are also a type of conic section, which means they can be created by slicing a cone at a certain angle. They also appear when we graph equations that involve squaring a number, like y = ax² + bx + c. These graphs always form a parabola shape.

Parabolas have useful properties. If made of something that reflects light or sound, rays or waves coming in parallel will bounce off and meet at the focus point. This property is used in tools such as parabolic antennas and headlights. Parabolas are important in physics and engineering.

History

The earliest known work on curves called parabolas was done by Menaechmus in the 4th century BC. He used parabolas to solve a special math problem. Later, Archimedes calculated the area inside a parabola and a line using a clever method called the "method of exhaustion."

The name "parabola" comes from the Greek word meaning "application." Galileo discovered that objects flying through the air, like balls, follow a parabolic path because of gravity. Today, parabolic shapes are used in satellite dishes, radar, and many types of telescopes because they help focus signals and light.

Definition as a locus of points

A parabola is a special curve in math. Every point on the parabola is the same distance from a fixed point called the focus and a fixed line called the directrix. Imagine you have a point (the focus) and a line (the directrix). Each point on the parabola is the same distance from the focus as it is from the directrix.

The point where the distance from the focus to the directrix is smallest is called the vertex. The line that runs through the focus and the vertex is the axis of symmetry. This means the parabola is perfectly mirrored along this line.

In a Cartesian coordinate system

In Cartesian coordinates, a parabola is a special curved shape. It can be described using a point called the focus and a line called the directrix.

When the tip of the parabola, called the vertex, is at the origin and the directrix is the line ( y = -f ), the focus is at ( (0, f) ). A point ( P = (x, y) ) is on the parabola if the distance from ( P ) to the focus is the same as the distance from ( P ) to the directrix.

This creates a U-shaped curve that opens upwards. The equation for this parabola is ( y = \frac{1}{4f}x^{2} ).

For a parabola with a vertex at ( V = (v_{1}, v_{2}) ), the equation changes to match the position of the vertex. The shape stays U-shaped, and the focus and directrix move to match the vertex's location.

As a graph of a function

A parabola can be drawn by using a simple math rule: f(x) = ax², where a is any number except zero. If a is positive, the parabola opens upward like a U-shape. If a is negative, it opens downward like an upside-down U.

All parabolas are related to each other. You can move, turn, or resize a parabola, and it will still be a parabola. This means any parabola can be changed to look like the basic one y = x² by moving and stretching it just right.

As a special conic section

A parabola is a special type of shape called a conic section. It is one of a group of shapes that follow the same rules. All these shapes have a middle point and are symmetric along a line.

We can change a special number called the eccentricity to see how the shape looks. If this number is zero, the shape becomes a circle. If it is exactly one, the shape becomes a parabola. If the number is bigger than one, the shape becomes a hyperbola.

In polar coordinates

A parabola can also be described using polar coordinates. When the equation is ( y^{2} = 2px ) and ( p > 0 ), the parabola opening to the right can be written in polar form. The vertex of this parabola is at the origin ((0,0)), and its focus is at ((\frac{p}{2},0)).

If we move the origin to the focus, the equation changes to a different polar form. This shows that a parabola has special properties when studied using polar coordinates.

Conic section and quadratic form

The diagram shows a cone with its axis AV and apex A. An inclined cross-section of the cone, shown in pink, forms a parabola. This parabola is the edge of the cross-section when viewed from the side.

A parabola can also be described using a math rule. By looking at the relationships between different points and lines in the cone's cross-section, we can find a rule that shows how the points of the parabola are related. This rule helps us understand the shape and features of the parabola better.

The parabola's shape comes from the way it curves, and this can be shown through simple math relationships between distances.

Proof of the reflective property

The reflective property of a parabola means that light traveling parallel to the axis of symmetry will reflect toward the focus. This idea comes from how light behaves, traveling in straight lines called rays.

To understand this, think about a simple parabola shaped like y = x². No matter how a parabola is shaped, this example helps explain how they all work. When light hits the parabola at any point, it reflects in a special way toward the focus, a specific point inside the curve. This special behavior happens because of the way the parabola is shaped.

Pin and string construction

You can draw a parabola using just pins and a string! First, pick a point called the focus and a line called the directrix. Then, take a triangle and a string of a fixed length. Pin one end of the string to the focus and the other end to a point on the triangle. Slide the triangle along the directrix while keeping the string tight with a pen. As you move, the pen will trace out a curve called a parabola. This works because the distance from any point on the curve to the focus equals the distance to the directrix.

Properties related to Pascal's theorem

A parabola is a special kind of curve with unique properties linked to a mathematical idea called Pascal's theorem. These properties help us see how points and lines connect to the parabola.

One good way to think about a parabola is by using a simple equation, like ( y = x^2 ). This makes it easier to study its patterns. For example, if you know four points on the parabola, you can find that some lines will always be parallel to each other. This pattern works no matter which four points you pick on the curve.

There are also cool tricks with tangents—lines that touch the parabola at just one point. If you have three points on the parabola and know the tangent at one of them, you can find the tangent at another point with a few simple steps. These ideas help mathematicians and scientists solve problems and draw accurate pictures of parabolas.

Steiner generation

Steiner described a way to build points on a special U-shaped curve called a parabola. This method uses two sets of lines meeting at points U and V, with a special mapping between them. The points where matching lines cross each other create the parabola.

For the parabola shaped like ( y = ax^2 ), we start at the bottom point S (0, 0). We pick a point P on the parabola and connect it to points on the axes. By dividing and projecting these connections, we can find more points that lie on the parabola. This shows how the parabola can be built step by step.

The same idea can also help create a "dual parabola," which is made from the lines that just touch a normal parabola. By using three points and connecting them in a certain way, we can find these touching lines, which are parts of the dual parabola. This method links to a special kind of curve called a Bézier curve.

Main article: Steiner conic

Remark: Steiner's method also works for other curves like ellipses and hyperbolas.

ellipses

hyperbolas

Bézier curve

de Casteljau algorithm

Inscribed angles and the 3-point form

A parabola can be described by an equation of the form ( y = ax^2 + bx + c ), where ( a \neq 0 ). This equation is found using three points with different x-coordinates. By using the coordinates of these points in the equation, we get three new equations. Solving these helps find the values of ( a ), ( b ), and ( c ).

There is a special rule for parabolas, much like the inscribed angle theorem for circles. For four points on a parabola, the angles at two of the points can be equal under certain conditions. This rule helps us understand parabolas better and find their equation using just three points.

Pole–polar relation

A parabola can be described by a simple math rule: y = a x². This rule helps us understand special connections between points and lines related to the parabola.

There is a special link called the pole–polar relation. For any point on the parabola, the matching line is the tangent line that just touches the parabola at that point. For points outside the parabola, the matching line connects to two tangent lines from that point. For points inside the parabola, the matching line does not touch the parabola at all. These relationships help mathematicians study parabolas and other shapes like ellipses and hyperbolas.

Main article: pole–polar relation

Tangent properties

A parabola has special properties related to lines that just touch it, called tangents. One property is that if two tangents are at right angles to each other, they will meet at a specific line called the directrix.

Another important rule is Lambert's theorem. If three tangents form a triangle, the focus of the parabola — a special point inside it — will always lie on the circle that passes through all three corners of that triangle.

Main article: Orthoptic (geometry)

Facts related to chords and arcs

When a line called a chord crosses a parabola at right angles to its middle line, or axis, there are special ways to find the distance to the point called the focus. This distance is called the focal length. We can find it using the length of the chord and how far the chord is from the vertex, the point where the parabola changes direction.

There is also a special rule for finding the space, or area, between a parabola and a chord. This area is always two-thirds of the space inside a shape called a parallelogram, which has the chord as one of its sides. This idea was first discovered a long time ago by a mathematician named Archimedes.

These rules help us understand how parabolas behave and can be used in real-life objects like satellite dishes and curved mirrors.

A geometrical construction to find a sector area

This section explains a special way to find the area of a part of a parabola. A parabola is a curved shape that looks like a U.

The method was created by Isaac Newton. You can read about it in his famous book, Philosophiæ Naturalis Principia Mathematica.

To find this area, we use a point called the focus (S) and a line called the directrix. We also need the main point of the parabola (V). By drawing certain lines and measuring distances, we can figure out the area of a slice of the parabola. This helps us understand how shapes and motion are connected.

Focal length and radius of curvature at the vertex

The focal length of a parabola is half of its radius of curvature at its vertex. If you know the focal length, you can find the radius of curvature by doubling it.

This helps explain why small parts of a spherical mirror can act like a parabolic mirror, focusing light to a point. The measurements in these relationships are often given in units related to the latus rectum, which is four times the focal length.

As the affine image of the unit parabola

Any parabola can be made from a basic parabola. This basic parabola is described by the simple equation y = x². We can move, stretch, or turn this basic shape to create any other parabola. Using special math steps called affine transformations, we can start with the unit parabola and change it in many ways.

These steps let us slide the parabola to a new place, change which way it points, and stretch or squeeze it. Even though the math looks tricky, it simply means we can take the basic U-shape and move or reshape it to make any parabola we can imagine.

As quadratic Bézier curve

A quadratic Bézier curve is a special kind of curve. It is defined by three points, called control points. These points guide the shape of the curve. The curve is part of a parabola. It changes smoothly between the points, creating a smooth, curved path.

Numerical integration

One way to find the area under a curve is to use pieces of parabolas. A parabola looks like a U and can be described using three points. This method is called Simpson's rule. It helps us guess the area under a curve by using the shape of parabolas.

As plane section of quadric

Some special 3D shapes, called quadrics, can show parabolas when you look at them from certain flat views. These shapes include the elliptical cone, parabolic cylinder, elliptical paraboloid, hyperbolic paraboloid, hyperboloid of one sheet, and hyperboloid of two sheets. Each of these shapes can display a parabola when cut in the right way.

As trisectrix

A parabola can help divide an angle into three equal parts using just a straightedge and compass. Normally this is very hard to do with only these tools.

This way of doing it was first written about by René Descartes in his book La Géométrie in 1637. It uses the special shape of the parabola to find the exact answer.

Generalizations

When we think about parabolas using different kinds of numbers, many of their basic shapes stay the same. For example, a straight line will still meet a parabola in at most two points.

In more advanced math, parabolas can be used in higher dimensions. One way is through objects called rational normal curves, where the standard parabola is like the simplest case. There are also shapes like the elliptic paraboloid and hyperbolic paraboloid, which are like parabolas stretched in three dimensions.

Main article: rational normal curves Main articles: elliptic paraboloid, hyperbolic paraboloid

In the physical world

Parabolas appear in many places in nature. One example is the path a ball follows when thrown into the air. This path is called a parabolic trajectory. It was first studied by Galileo. Air can change the exact shape, but the basic path looks like a parabola.

Parabolas are also used in technology and buildings. The curved cables of suspension bridges often have a parabolic shape. Parabolic mirrors and antennas focus light or radio waves to one point. This is useful for telescopes, satellite dishes, and solar cookers. When a liquid spins inside a container, its surface forms a parabolic shape.