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I ploted 3D the function Sin[A/2]Sin[B/2]Sin[C/2] with A, B, C > 0 and A + B + C = Pi.

A basic approach in the post How to plot ternary density plots answers with the use of FindGeometricTransform. How can I transform the Plot3Ded function inside the equilateral triangle with FindGeometricTransform? If there is a simpler method, I would like to know it.

An example of ternary 3D plot of a geometric inequality

Let ABC be a triangle and its radius of circumcircle R = 1.

In ternary 3D plot, based on A, B, C > 0 and A + B + C = Pi, the value of a function is plotted 3-dimensionally on the equilateral domain.

Denote:

a, b, c = the sidelengths of ABC respectively.

s = (a + b + c)/2.

r = the radii of incenter.

ha, hb, hc = the altitudes of ABC respectively.

The bottom (yellow), middle (magenta) and top (cyan) surfaces show the values of left-hand-side, middle and right-hand-side functions of the following inequality by Mr. George Apostolopouls.

Ga19Mar30: 6rs/R ≤ Sin[A](hb + hc) + Sin[B](hc + ha) + Sin[C](ha + ha) ≤ 3s. enter image description here

I ploted 3D the function Sin[A/2]Sin[B/2]Sin[C/2] with A, B, C > 0 and A + B + C = Pi.

A basic approach in the post How to plot ternary density plots answers with the use of FindGeometricTransform. How can I transform the Plot3Ded function inside the equilateral triangle with FindGeometricTransform? If there is a simpler method, I would like to know it.

I ploted 3D the function Sin[A/2]Sin[B/2]Sin[C/2] with A, B, C > 0 and A + B + C = Pi.

A basic approach in the post How to plot ternary density plots answers with the use of FindGeometricTransform. How can I transform the Plot3Ded function inside the equilateral triangle with FindGeometricTransform? If there is a simpler method, I would like to know it.

An example of ternary 3D plot of a geometric inequality

Let ABC be a triangle and its radius of circumcircle R = 1.

In ternary 3D plot, based on A, B, C > 0 and A + B + C = Pi, the value of a function is plotted 3-dimensionally on the equilateral domain.

Denote:

a, b, c = the sidelengths of ABC respectively.

s = (a + b + c)/2.

r = the radii of incenter.

ha, hb, hc = the altitudes of ABC respectively.

The bottom (yellow), middle (magenta) and top (cyan) surfaces show the values of left-hand-side, middle and right-hand-side functions of the following inequality by Mr. George Apostolopouls.

Ga19Mar30: 6rs/R ≤ Sin[A](hb + hc) + Sin[B](hc + ha) + Sin[C](ha + ha) ≤ 3s. enter image description here

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xzczd ♦
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I ploted 3D the function Sin[A/2]Sin[B/2]Sin[C/2]Sin[A/2]Sin[B/2]Sin[C/2] with A, B, C > 0A, B, C > 0 and A + B + C = PiA + B + C = Pi. A basic approach in the title "How to plot ternary density plots"

A basic approach in the post How to plot ternary density plots answers with the use of "FindGeometricTransform"FindGeometricTransform. How can I transform the Plot3DedPlot3Ded function inside the equilateral triangle with "FindGeometricTransform"FindGeometricTransform? If there iais a simpler method, I would like to know it.

I ploted 3D the function Sin[A/2]Sin[B/2]Sin[C/2] with A, B, C > 0 and A + B + C = Pi. A basic approach in the title "How to plot ternary density plots" answers with the use of "FindGeometricTransform". How can I transform the Plot3Ded function inside the equilateral triangle with "FindGeometricTransform"? If there ia a simpler method, I would like to know it.

I ploted 3D the function Sin[A/2]Sin[B/2]Sin[C/2] with A, B, C > 0 and A + B + C = Pi.

A basic approach in the post How to plot ternary density plots answers with the use of FindGeometricTransform. How can I transform the Plot3Ded function inside the equilateral triangle with FindGeometricTransform? If there is a simpler method, I would like to know it.

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How to ternary Plot3D a function

I ploted 3D the function Sin[A/2]Sin[B/2]Sin[C/2] with A, B, C > 0 and A + B + C = Pi. A basic approach in the title "How to plot ternary density plots" answers with the use of "FindGeometricTransform". How can I transform the Plot3Ded function inside the equilateral triangle with "FindGeometricTransform"? If there ia a simpler method, I would like to know it.