Interactive 3D visualisation of two intersecting cones

Builds an interactive plotly widget showing two cones with common apex on the unit sphere, their intersection circle and intersection points, the spherical excess region, and the auxiliary angles (\(\alpha, \beta, \gamma\)) that appear in the Mazonka (2012) analytic formula for the solid angle of the intersection.

Usage

plotIntersectingCones(
  theta1 = pi/6,
  theta2 = pi/4,
  phi = pi/3,
  radius = 1,
  color_palette = "viridis",
  show_rays = FALSE,
  cone_opacity = 0.2
)

Arguments

theta1

Apex angle of first cone in radians (default: pi/6)

theta2

Apex angle of second cone in radians (default: pi/4)

phi

Separation angle between cone axes in radians (default: pi/3)

radius

Sphere radius (default: 1)

color_palette

Color palette name: "viridis", "plasma", "inferno", "magma", "cividis", "rainbow" (default: "viridis")

show_rays

Logical; if TRUE, show cone edge rays (default: FALSE)

cone_opacity

Opacity of cone surfaces, range $[0, 1]$ (default: 0.2)

Value

A plotly object with interactive 3D visualization

Details

The visualization shows two cones with apex angles \(\theta_1\) and \(\theta_2\), separated by angle \(\phi\) between cone axes; intersection points on the unit sphere are marked, along with great circle arcs and planar circle arcs connecting them. The display includes angles \(\alpha\), \(\beta\), and \(\gamma\) relevant to solid angle computation, providing a complete geometric view of the intersecting cone configuration.

References

Mazonka, O. (2012). Solid angle of conical surfaces, polyhedral cones, and intersecting spherical caps. arXiv:1205.1396 (math.MG). doi:10.48550/arXiv.1205.1396

Sievert, C. (2020). Interactive Web-Based Data Visualization with R, plotly, and Shiny. Chapman and Hall/CRC. ISBN 978-1138331457.

See also

solid_angle_intersecting_cones for the closed-form measure of the region shown by the visualisation; plot_cone_3d for the simplicial-cone visualisation; shinySolidAngleR for the dashboard that embeds these widgets.

Examples

if (FALSE) { # \dontrun{
# Default configuration with semi-transparent cones
plotIntersectingCones(color_palette = "viridis")

# Custom angles and palette
plotIntersectingCones(theta1 = pi/4, theta2 = pi/3, phi = pi/5,
                      color_palette = "magma", cone_opacity = 0.25)
} # }