The input in yellow goes directly through the origin. The polynomial in red never makes it to positive time. Near the origin, a second red dot appears to run at the other for an annihilation, which should become a familiar sight. Because the coefficients are reals, and the input goes through the origin, the polynomial is in exactly the same line as the input. There is nothing to "knock it" into another place in space.

The input which travels through the origin is shown in yellow. A cubic polynomial is in red. The polynomial continues its march down and to the left after a brief moment of indecision. The coefficients are all real, so the red and yellow are all in the same line.

The input is in yellow and goes through the origin. The polynomial in red has real coefficients. Perhaps a different polynomial could switch directions 5 times, although I have yet to construct one.

The yellow line is the input, the red line a 2nd order polynomial with real coefficients, and the green line is the same polynomial with quaternion coefficients. The coefficients are only subtly different as shown in the animation. The green line is never quite in line with the yellow/red line.

The input in yellow passes through the origin. The red input has real coefficients, and stays exactly in line with the yellow. The green line with its complex coefficients curves though space, but has the same large scale features as the red line - a reverse s shape.

The yellow line is the input that goes through the origin for red, a cubic polynomial with real coefficients. The blue line is an input that does not go through the origin, making the curved orange line. That curve appears to line in a plane defined by the yellow and blue lines. The shape of the orange line is distinct from that of the red line. What goes in can dramatically change what comes out. The sharp bend in the t-x graph for the orange line was unexpected.