Hartmut F. W. Höft
The sickle of the moon can be drawn with two ellipses in the plane and
a pair of sickles with two orthogonal ellipses. I asked myself what
other designs I might be able to create with such pairs as the only
geometric element in the plane. With my images I want to demonstrate
to the viewer the conundrum of the easy and the difficult, the simple
and the complex. The images are based on two fundamental geometric
elements: a curve in the plane that provides the "spine" of a figure
in an image and pairs of orthogonal ellipses that are placed at
regular intervals along the spine and that create the shape in the
image. The spine is not drawn, only the ellipses are drawn. I render
all images in Mathematica.
What surprised me most, however, is that many of the resulting images appear three-dimensional. Our visual system apparently gets fooled into seeing a three-dimensional object when we draw many ever smaller slices after larger ones in a single plane. All images are mathematically planar.
In this image 720 pairs of orthogonal ellipses are drawn in the plane along a closed planar curve whose loop structure has intrigued me for some time. The ratio of the semi-axes in one ellipse is 1.05 which is close to the ratio of the equatorial and polar radii of earth and moon; the other ellipse has a ratio of 1.25. Starting colors for the ellipses are red and cyan. The lengths of the semi-axes are modified countercyclically by f(t) = (sin^2 t, cos^2 t) and g(t) = (2 sin^2 t, 2 cos^2 3t), respectively. The underlying spine is given by r = sin^3(5t) + cos^5(3 t) in polar coordinates. This is a closed curve with five asymmetric loops and lobes. One of the lobes appears as the red part of the body in the "background" while another is represented by the black spikes on the right side. Since the entire spine is traced twice, its dominant loop appears in a multi-colored double overlay in the lower left of the image. The graphics were rendered in Mathematica 8.
In this image 180 pairs of orthogonal ellipses are drawn in the plane along a closed curve that belongs to a family that I used in the classroom when teaching polar coordinates. The ratio of the semi-axes in the ellipses is 1.05 which is close to the ratio of the equatorial and polar radii of earth and moon. Starting colors for the ellipses are green and magenta. The lengths of the semi-axes are modified synchronously by the Gaussian Bell Curve f(t)=e^ -(t^2) over the interval [-1, 1]. The underlying spine is a section from -pi/2 to pi/2 of the curve r = sin^4 (4t) + cos (3 t) rotated clockwise by 90 degrees. This is a closed curve symmetric about the x-axis with three loops and one lobe on each side of the axis of symmetry. The ellipses are rotated a total of 3*pi, but not all of that rotation is visible because of the large differences in the scale of the loops and the semi-axes of the ellipses. The graphics were rendered in Mathematica 8.
In this image 36 pairs of orthogonal ellipses are drawn in the plane. This is one of the earliest images that I created and that at first sight appeared to me to be three-dimensional. It is an image that is only one step away from drawing a planar spiral with pairs of ellipses. The ratio of the semi-axes in the ellipses is 2.5 which makes large, paired portions of both ellipses visible rather than just paired slivers of one ellipse with the other ellipse dominating. Starting colors for the ellipses are black and white. The lengths of the semi-axes are shrunk linearly towards zero in steps. The pairs of ellipses are rotated two full turns in discrete steps while at the same time their center is moved along the horizontal axis to the right. The graphics were rendered in Mathematica 8.