已知密度D1、D2，如何推导求解高度与半径的比值H/R

Got it, let's work through this step by step. The key here is to use a substitution to simplify the ratio into a single-variable equation—since we only care about ( \frac{H}{R} ), we can eliminate the separate ( R ) and ( H ) terms entirely. (I’ll assume ( h=H ) and ( r=R ) to keep notation consistent; let me know if that’s not the intended alignment!)

First, start with your derived density ratio:

$$\frac{D2}{D1} = \frac{3Rh^2 - h^3}{4R3}$$

Step 1: Define a substitution variable #

Let ( x = \frac{H}{R} ). This means ( H = xR )—we can now rewrite every ( H ) in terms of ( x ) and ( R ).

Step 2: Substitute ( H = xR ) into the equation #

Plug ( H = xR ) into the numerator of the right-hand side:

• Numerator: ( 3R(xR)^2 - (xR)^3 = 3R \cdot x^2R2 - x^3R3 = R^3(3x2 - x^3) )
• Denominator: ( 4R^3 )

Cancel out ( R^3 ) from the numerator and denominator (since ( R \neq 0 )), and we get a simplified equation in terms of ( x ):
$$\frac{D2}{D1} = \frac{3x^2 - x^3}{4}$$

Step 3: Rearrange into a solvable cubic equation #

Let ( k = \frac{D2}{D1} ) (this just cleans up the notation). Multiply both sides by 4:
$$4k = 3x^2 - x^3$$

Rearrange into standard cubic form (all terms on one side):
$$x^3 - 3x^2 + 4k = 0$$

Step 4: Solve for ( x = \frac{H}{R} ) #

This is a cubic equation in ( x ). You have two main options here:

1. Closed-form solution: Use Cardano's formula for cubic equations. While this gives an exact answer, it’s algebraically messy for practical use.
2. Numerical solution: Since ( x = \frac{H}{R} ) is a positive real number (heights and radii are positive), numerical methods like Newton-Raphson iteration work well. For example:
   • Start with an initial guess ( x_0 ) (if this relates to a spherical cap/segment, ( x ) will be between 0 and 2, so ( x_0=1 ) is a safe starting point).
   • Iterate using:
     $$x_{n+1} = x_n - \frac{x_n^3 - 3x_n^2 + 4k}{3x_n^2 - 6x_n}$$
   • Stop when the change between ( x_{n+1} ) and ( x_n ) is smaller than your desired precision.

For example, if ( \frac{D2}{D1} = 0.5 ) (so ( k=0.5 )), the cubic equation becomes ( x^3 - 3x^2 + 2 = 0 ). Factoring gives ( (x-1)(x^2-2x-2)=0 ), with valid positive solutions ( x=1 ) (since ( x=1+\sqrt{3} \approx 2.732 ) would imply ( H > 2R ), which is likely impossible for the original shape).

内容的提问来源于stack exchange，提问作者Jake Stan