Model for Descriptive Geometry by A. Jullien - Representation of a Point

For the discussion that follows, the following conventions will be used to explain the location of points in each model. Any point in space can be denoted by a coordinate triple (x,y,z). This is the three-dimensional version of the (x,y) plane from Euclidian geometry learned in high school. For our purposes, the x-axis will be the horizontal line stretching left and right at the fold of each relief. The y-axis will be on the horizontal plane (paper card) of each model, appearing to be coming out of the plane of each image. The z-axis will be the vertical axis of each relief and lies on the vertical plane (paper card) of each relief. In all of the reliefs, the x-coordinate is irrelevant since each projection will be with respect to the y and z planes. When needed, a point will be referred to in two coordinates only (y,z), leaving off the x-coordinate for brevity. Positive will be the forward or upward direction and negative values will be behind or below the cards of each relief.

In each relief, a point of interest that is on the vertical or horizontal plane will be marked by a letter and a small hole or dot. Points in space are shown by the bend in a wire that pierces the cards at the y and z intercepts (a,0) and (0,b) respectively and will be denoted (a, b). Lines are shown by black or red strings threaded between points on the cards or by wires and will be denoted by the point on the horizontal plan followed by the point on the vertical plane, such as ab. The title of each relief is actually a construction. For example, relief seven is entitled “line perpendicular to a plane.” This is actually a task, “construct a line perpendicular to a given plane in space.” Following Jullien, we will assume the directions are to construct the item in question. There are many projections shown for the construction of each item. For simplicity, I have only described the relevant items in each model, leaving out all the mathematical details. It would take a whole textbook on the topic to rigorously go through each model. And that is the point of the models, to supplement the textbook Jullien wrote. The reliefs slowly progress from simple to complex, starting with the depiction of points and lines and ending with the construction of a pyramid, guiding the students through the constructions of descriptive geometry. The progression of the reliefs follows the textbook.

In this particular model, nine points are shown for all the possible combinations of a positive, negative or zero value for y or z. For example, the first point on the left shows a point with both y and z coordinates positive.

Reference:

A. Jullien, Cours élémentaire de géométrie descriptive. . ., Paris: Gauthier-Villars, 1875. There were editions of this book as late as 1887.

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