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(complex_3d)

Complex Roots in 3D

The roots x = −c ± √(c² − 1) of x² + 2cx + 1 = 0 trace two curves as c varies from −2 to 2. The three axes show the parameter c, the real part of x, and the imaginary part of x. Blue follows the plus branch; red follows the minus branch.

For |c| < 1, the roots are complex conjugates with real part −c and imaginary parts ±√(1 − c²). Outside that interval, both roots lie in the gridded Im(x) = 0 plane. The markers show the repeated roots at c = ±1, where the branches meet: x = 1 at c = −1, and x = −1 at c = 1.

Graph applies isometric_projection() to the curves, axes, ticks, and labels. Each SymLine samples a root as {x: c, y: Re(x), z: Im(x)}, keeping its real and imaginary components together throughout the curve. The graph's limits describe the projected view, and its aspect preserves equal scale in that view.

Compare Complex Roots, which plots the real and imaginary components separately, or see the projection guide.