探究仅含实根的二次多项式系数构成的3D有界区域形状

我一直以来都记得，二次多项式 $ax^2+bx+c$ 只有实根的充要条件是判别式 $D \geq 0$，具体可以拆成两种情况：
$$
D = b^2 - 4ac \geq 0 \Rightarrow
\begin{cases}
b^2 \geq 4ac & \text{if } ac > 0, \
\text{any } b & \text{if } ac \leq 0.
\end{cases}
$$

最近我琢磨着写个程序，能直接生成满足“仅含实根”条件的随机二次多项式系数，但不想用if语句做筛选——毕竟如果靠先随机生成再筛选的话，模拟时会扔掉大量不符合的样本，效率实在太低了。

最开始我用Mathematica写了一段筛选式的代码来观察这个区域的形状：

n = 10000; (* Number of random sets to generate *)
data = Table[
a = RandomReal[{-2, 2}];
b = RandomReal[{-2.5, 2.5}];
c = RandomReal[{-3, 3}];
discriminant = b^2 - 4*a*c;
If[discriminant >= 0, {a, b, c}, Nothing],
{n}
];
chm = ConcaveHullMesh[data];
bdp = MeshCoordinates[RegionBoundary[chm]];
ListSurfacePlot3D[bdp, AxesLabel -> {"a", "b", "c"}, PlotLabel -> "Coefficients of Quadratics with D >= 0"]

跑出来的3D形状有点意思：不管我调整a、b、c的取值范围，生成的区域都是上下部分平坦，中间带有曲线的形态，而且看起来明显有翻转对称性。

这就让我好奇了：给定a、b、c各自的最大最小值范围后，所有满足“二次多项式仅含实根”的点(a,b,c)构成的到底是什么形状？

更新：边界形态的多视角可视化 #

为了更清晰地观察这个区域的边界（也就是满足$b^2=4ac$的点），我又写了一段代码，专门生成边界上的随机点，并从多个正交视角进行可视化：

n = 10000; (* Number of random sets to generate *)
data = Table[
a = RandomReal[{-2, 2}];
c = RandomReal[{-3, 3}];
If[RandomChoice[{True, False}],
b = 2 * Sqrt[Abs[a * c]], (* Ensure b is real *)
b = -2 * Sqrt[Abs[a * c]]
];
If[a < 0, c = -Abs[c]]; (* Ensure c is negative if a is negative *)
If[c < 0, a = -Abs[a]]; (* Ensure a is negative if c is negative *)
{a, b, c},
{n}
];
viewPoints = {
{"Above", {0, 0, Infinity}}, {"Below", {0, 0, -Infinity}},
{"Front", {0, Infinity, 0}}, {"Behind", {0, -Infinity, 0}},
{"Left", {Infinity, 0, 0}}, {"Right", {-Infinity, 0, 0}},
{"Top Front Right", {Infinity, Infinity, Infinity}}, {"Bottom Back Left", {-Infinity, -Infinity, -Infinity}}
};
plots = Table[
ListPlot3D[data,
ViewPoint -> vp[[2]],
ViewProjection -> "Orthographic",
ImagePadding -> 5,
PlotLabel -> Style[vp[[1]], 14, Bold, Background -> White],
PlotRangePadding -> 0.2
],
{vp, viewPoints}
];
(* Adjust the size of the graphics grid as needed *)
Rasterize[GraphicsGrid[Partition[plots, 2], Spacings -> {0, 0}, ImageSize -> Full]]

这段代码生成了多视角的可视化图，能更直观地看到边界的具体形态。

备注：内容来源于stack exchange，提问作者Romogi