Name: plotting points with polar coordinates
Uploaded: 2025-11-26T05:44:19.000Z
Duration: 1 min 24 s
Channel: JoVE

Considere un dron volando en el plano xy.Para localizar su posición, se necesitan dos mediciones.La primera es la distancia desde un punto central, generalmente el origen. El segundo es el ángulo que forma esta línea con el eje x positivo. Juntos, estos dos valores definen la ubicación del dron en coordenadas polares.En este sistema, el punto en el centro se llama polo y el rayo horizontal del polo es el eje polar.El ángulo se mide a partir del eje polar. Los ángulos en sentido contrario a las agujas del reloj son positivos, mientras que los del sentido de las agujas del reloj son negativos.Teniendo en cuenta un punto de referencia en el plano de coordenadas polares, si el dron se encuentra en la dirección opuesta a este punto, se supone que el radio adquiere un valor negativo. Dado que agregar múltiplos de 2π no cambia la ubicación, un punto tiene infinitas representaciones polares.Las coordenadas polares y cartesianas están vinculadas por funciones trigonométricas. Las coordenadas x e y se determinan multiplicando el radio por el coseno y el seno del ángulo. Por el contrario, el radio se obtiene del teorema de Pitágoras y el ángulo del arco tangente de y sobre x.

coordenadas polares

Polar Coordinates

El sistema de coordenadas polares ofrece una alternativa al sistema de coordenadas cartesianas para especificar puntos en un plano, mediante una distancia y un &#225;ngulo en lugar de las coordenadas x e y. Este sistema es especialmente ventajoso en situaciones con simetr&#237;a circular o rotacional, como en problemas de f&#237;sica o ingenier&#237;a que implican ondas, oscilaciones o trayectorias orbitales.Definici&#243;n de coordenadas polaresEn coordenadas polares, un punto se representa como P(r, &#952;), donde r es la distancia radial desde un punto fijo llamado polo (an&#225;logo al origen en coordenadas cartesianas) y &#952; es el desplazamiento angular desde una direcci&#243;n fija conocida como eje polar (habitualmente el eje x positivo). El &#225;ngulo &#952; se mide en radianes; es positivo cuando se mide en sentido antihorario y negativo cuando se mide en sentido horario. Cuando r es negativo, el punto se encuentra sobre la recta en la direcci&#243;n opuesta al &#225;ngulo &#952;.Representaciones m&#250;ltiples y equivalencia de coordenadasUna caracter&#237;stica importante de las coordenadas polares es que un mismo punto puede representarse de varias maneras. Dado que los &#225;ngulos que difieren en m&#250;ltiplos enteros de 2&#960; radianes determinan la misma direcci&#243;n, el punto P(r, &#952;) es equivalente a P(r, &#952; + 2n&#960;) para cualquier entero n. Adem&#225;s, se puede representar el mismo punto empleando un radio negativo como P(&#8722;r, &#952; + &#960;).Conversi&#243;n entre sistemas de coordenadasPara convertir entre coordenadas polares y cartesianas se aplican relaciones trigonom&#233;tricas:De polar a cartesiana:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:54.9%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/40081/40081_Equation_1_editor_40081.svg' alt='40081_Equation_1.svg' /></figure>De cartesiana a polar:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:54.32%;'><img style='aspect-ratio:1920/250;' src='https://cloudfront.jove.com/files/ftp_upload/40081/40081_Equation_2_editor_40081.svg' alt='40081_Equation_2.svg' /></figure>Estas relaciones facilitan la conversi&#243;n entre sistemas de coordenadas seg&#250;n el contexto del problema o la configuraci&#243;n geom&#233;trica.

Coordenadas polares

El sistema de coordenadas polares ofrece una alternativa al sistema de coordenadas cartesianas para especificar puntos en un plano, mediante una distancia ...

Education

Educación

JoVE Core

Math Fundamentals

Fundamentos de matemáticas

Unit

Unidad

Polar Coordinates and Vectors

Coordenadas polares y vectores

KEY TERMS AND CONCEPTS

TÉRMINOS Y CONCEPTOS CLAVE

plotting points with polar coordinates

Polar coordinates give a way to plot points in a plane using distance and angle. Instead of x and y values, this system uses a radial distance and an angular direction. It is especially useful for problems with circular or rotational symmetry, including waves, oscillations, and orbital paths in physics and engineering.

A point in polar form is written as P(r, θ). The value r is the distance from the pole, which plays the same role as the origin in the Cartesian plane. The angle θ is measured from the polar axis, usually the positive x-axis, and it is measured in radians. Positive angles move counterclockwise, while negative angles move clockwise.

A negative radius changes the direction of the point. When r is negative, the point is placed on the line opposite the angle θ. Polar coordinates can also represent the same point in more than one way. Angles that differ by 2nπ radians, where n is any integer, point in the same direction. The same point can also be written with a negative radius as P(-r, θ + π).

Polar and Cartesian coordinates can be converted using trigonometric relationships. To move from polar to Cartesian form, the coordinate values are found from r and θ. To move from Cartesian to polar form, the same relationships are used in reverse. These conversions make it easier to choose the coordinate system that best fits the shape or situation being studied.

Plotting Points with Polar Coordinates

polar coordinates

Polar coordinates give a way to plot points in a plane using distance and angle. Instead of x and y values, this system uses a radial distance and an angular direction. It is especially useful for problems with circular or rotational symmetry, including waves, oscillations, and orbital paths in physics and engineering. A point in polar form is written as P(r, θ). The value r is the distance from the pole, which plays the same role as the origin in the Cartesian plane. The angle θ is measured...

graphs of polar equations

Polar Curves: Circles, Roses, and Limacons

Polar equations can be used to graph curves with radial symmetry or repeating patterns. In the polar coordinate system, a point is described by its distance from the pole, or central point, and by an angle from the polar axis, or reference direction. This format is different from rectangular coordinates and is especially useful for certain curve shapes. A circle centered at the pole appears when the radius stays constant as the angle changes. A line through the pole appears when the angle...

vectors

Vector Components and Direction

Vectors describe quantities with both magnitude and direction. This makes them different from scalars, which have magnitude only. Common examples include displacement, velocity, and force, where direction matters. A vector is shown as a directed line segment from an initial point to a terminal point. It may also be written with bold letters or with an arrow above the symbol. Two vectors are equal if they have the same magnitude and direction, even when they start in different places. Vector...

the dot product

Dot Product From Direction and Angle

The dot product measures how much one vector acts along another vector’s direction. Vectors are quantities with both size and direction. This calculation gives one number that shows how aligned two vectors are. One way to find the dot product is to combine the horizontal parts and the vertical parts of the two vectors. The matching parts are multiplied, and the results are added together. This form shows why the value depends on both direction and strength. The dot product becomes larger when...

key terms and concepts

Learn how polar coordinates use distance and angle to locate points, handle negative radii, and convert between polar and Cartesian forms.

Plotting Points with Polar Coordinates | JoVE K12

The JoVE Math Fundamentals collection is designed to build a deep and comprehensive understanding of essential mathematics. Covering the foundations of algebra, geometry, trigonometry, and analytic reasoning, this course establishes a strong base for future studies. The collection connects core mathematical concepts with real-world practice, equipping learners to solve complex problems, master problem-solving skills, and apply abstract ideas.

math fundamentals

Plotting Points with Polar Coordinates

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