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Cone plots (also known as 3-D quiver plots) represent vector fields defined in some region of the 3-D space.

A vector field associates to each point of coordinates (x, y, z) a vector of components (u, v, w).

In this post, we’ll explore how Plotly’s cone plots can be used to visualize atmospheric wind 💨, magnetic fields, a trajectory of the Rössler System, and tangent vector fields to a surface.

## So what’s a 3-D vector?

∙ A vector is a geometric object that has magnitude and direction.

∙ A 3-D vector is useful in any physical space where both magnitude and direction matter.

∙ In 3-D space, vectors are identified with triples of scalar components. In the case of the example image below: *a = (ax, ay, az)*

The direction of the vector field at a point is illustrated by a geometric cone, colored according to the vector norm (magnitude) at that point.

Cone plots can be used in a variety of abstract ways: to model a vortex (such as a tornado), illustrate an explicitly defined vector field, chart a hyperboloid, plot a system of non-linear ordinary differential equations, and show winds at various atmospheric levels.

For extra insights, check out our tutorial about making 3-D cone plots in Python with Plotly.

## Cone Plot Showing Atmospheric Wind

This plot uses an explicitly defined vector field. A vector field refers to an assignment of a vector to each point in a subset of space.

In this plot, we visualize a collection of arrows that simply model the wind speed and direction at various levels of the atmosphere.

3-D weather plots can be useful to research scientists to gain a better understanding of the atmospheric profile, such as during the prediction of severe weather events like tornadoes and hurricanes.

Speaking of tornado-like rotations, here is an example of a cone plot of a vortex. The size of the cones are driven by wind speed.

Python code | Data | Link to plot

## Cone Plot of a Magnetic Field

The Biot-Savart Law is an equation that describes the magnetic field created by a current-carrying wire.

It was named after Jean-Baptiste Biot and Felix Savart in 1820 when the pair derived the mathematical expression for magnetic flux density.

In this example, we leverage the law to create a hypothetical magnetic field created by an electric current moving through three circular loops.

## Cone Plot Showing a Trajectory of the Rössler System

The Rössler system is a system of three ordinary differential equations, whose dynamics exhibits an attractor, or a set of states, invariant under the dynamics, towards which neighboring states in a given basin of attraction asymptotically approach in the course of dynamic evolution.

Originally studied by Otto Rössler, these differential equations “define a continuous-time dynamical system that exhibits chaotic dynamics associated with the fractal properties of the attractor.”

The equations have been found to be useful in modeling equilibrium in chemical reactions.

## Cone Plot of a Tangent Vector Field

In mathematics, a tangent vector is a vector that is tangent to (or just touches) a curve or surface at a given point.

Tangent vector fields are an “essential ingredient in controlling appearance for applications ranging from anisotropic shading to texture synthesis and non-photorealistic rendering.”

Below, we examine the tangent vector field along the coordinate lines of a parameterized hyperboloid.

Python code | Data | Link to plot

With 3-D cone plots in Plotly, you can allow the user to visualize vector fields across a variety of disciplines: meteorology, engineering, mathematics, and more. Bring your vectors to life with cone plots in Plotly!

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