Conic sections: moiré

All the conic sections (ellipse, parabola and hyper­bola) can be obtained as moiré — an addi­tional geometric pattern that is formed by two over­laying images.

Take a trans­parency film and print straight stripes at a fixed distance between adja­cent ones. On another film, print circular stripes (concen­tric circles) of the same width and with the same distance between adja­cent ones.

If we overlay concen­tric circles with straight lines, one can see a family of parabolas. Every parabola is a chain of oppo­site vertices of curvi­linear quadri­lat­erals, bounded by adja­cent lines and adja­cent circles. Shifting the sheets, one can see para­me­ters of the parabola family changing and ponder the posi­tion of the direc­trix and the focus on the basis of the parabola defi­n­i­tion.

If one over­lays two iden­tical “сir­cular” sheets, one can see ellipses and inter­secting them hyper­bolas with common focuses which are the centres of circles. And in this case, too, shifting the sheets helps to spot these curves and track the curves’ para­me­ters change.