In this article we will show how to visualize the intersection of a portion of cone with a plan if this intersection is an ellipse. For this example, you must use version 3.4 of MathGraph32 (version 3.4.1 or later advised). You can download the last version on [this page> 65].
Let us start by creating a virgin figure with the menu File- New figure without unity length and then use menu item Constructions - Implement predefined construction. Let us open the directory Space Version2 and choose to implement construction named Frame In Space With Cone. A figure appears with a frame of space. You can in the top right corner of the figure capture the fat points allowing to make turn the frame around the axis (Oz) or to change the slope of this axis.
In the color palette, activate the red color and create with tool two points liked to the point locus which represents the basic circle of the cone (ellipse) and name these two points E and F (icon ).
Activate the continuous and thick line style. With icon create segments [SE] and [SF] then use icon to create two points linked to these two segments which we will call B and C. With tool create also a point related to the half-line [OK) which we will call D.
In the color palette, activate the color chestnut and in the line stye palette the thin dotted line style. Use icon to create polygon BCD (right click to create the polygon after the click on the third point) then icon to fill this polygon with a surface.
In the color palette, activate the magenta color. Use icon to create another point linked th the point locus representing the basic circle of the cone which we will call M.
Create with icon the segment [SM] which will be regarded as a generator of the cone. If necessary, use tool of capture so that the figure resembles the figure below. Our goal is to create the point of intersection of plan (IJK) with segment [SM].
Activate the black color and with icon create lines (OE), (OF), (DB) and (CD). Using tool of intersection now create the point P of intersection of lines (DB) and (OE) then the point Q of intersection of the lines (CD) and (OF).
With tool now create line (PQ) which is the line of intersection of plan (IJK) with the plan containing the basic circle of the cone.
In the same way create lines (OM) and (PQ) then their point of intersection which we will call R. The line (DR) is the line of intersection of plan (BCD) with plan (SOM). If necessary, move the point M so that the (DR) is secant with segment [SM]. With the icon create the point of intersection of line (DR) with segment [SM] which we will call N. The point N is then the point of intersection of the segment [SM] with plan (BCD). Here below the figure obtained.
We now will create the locus of point N generated by the positions of M on the point locus it is linked to.
In the color palette, activate the blue color. Click on icon of creation of a locus of points generated by a linked point. Click on point N traces of which will generate the locus then on point M positions of which will generate the locus. In the dialog box which appears, ask 600 points and validate.
Now let us fill this locus with a surface by using icon .
Our figure is finished. You can now mask (tool ) the lines and the points used for the construction and you get the figure below.