Appendi: D Sketches Contemporar Calculus Appendi: Sketching Planes and Conics in the XYZ Coordinate Sstem Some mathematicians draw horrible sketches of dimensional objects and the still lead productive, happ lives. Some mathematics students are terrible D artists and still get A grades. But it reall takes ver little time and practice to do decent sketches of planes and conic sections in D, and it can be satisfing to have the curve ou know is an ellipse actuall look like an ellipse. This Appendi illustrates some step b step was to draw the basic building blocks for man D shapes: the ais sstem, planes and conic sections. First Step: Invest in a pencil and a small, inepensive, clear plastic 0 60 90 o triangle with a centimeter scale (Fig. ). The make it much easier to create good D sketches. 60 o Man of the directions in this Appendi suggest drawing "help lines" to assist ou in putting the various objects in the appropriate positions and orientations. The final drawings often look better if these "help lines" are removed when Fig. the sketch is finished, so a pencil is better than a pen. Man of the directions require ou to draw lines parallel to the aes and to draw parallelograms, and a clear plastic straightedge (or a triangle) is ver useful for this. Finall, the centimeter scale produces small sketches appropriate for personal work. positive XYZ The following directions are for "b hand" sketches, ais sstem but the can also help ou when using CAD 0 o 0 o (computer aided design) and Draw computer programs. 0 o using the triangle to draw the and aes A. The XYZ ais sstem everthing starts here! Including an ais sstem in a D sketch gives 0 o Fig. the viewer an orientation and a perspective for the location of an object. () Draw a vertical line segment. () Sketch other line segments making 0 o angles with the vertical segment. The 0 60 90 o triangle is useful here (Fig. ). () If an of the variables take negative values, ou can etend the appropriate ais to the negative values using dashed lines. (4) If a scale is needed, put "tick" marks along the aes at cm intervals. Fig. (Fig. ) Since the coordinate sstem is needed in so man sketches, ou should eventuall get used to sketching it without the aid of a ruler.
Appendi: D Sketches Contemporar Calculus B. The XY, XZ, and YZ coordinate planes. (Actuall, rectangular pieces of the coordinate planes.) The following rectangular pieces of the coordinate planes are useful b themselves, and the are ver important aids for drawing curves and conics later. Kes: Start with an ais sstem. Use lines parallel to each ais. The result should be a parallelogram. Piece of the XY coordinate plane: (Fig. 4) () Pick a point on the ais and draw a line segment to the left of and parallel to the ais. () From the end of the segment in step (), draw a line to the ais that is parallel to the ais. Piece of the XZ coordinate plane: (Fig. 5) () Pick a point on the ais and draw a vertical line segment (up and parallel to the ais). () From the end of the segment in step (), draw a line to the ais that is parallel to the ais. Plane (piece) parallel to the plane: (Fig. 6) () Locate and label the appropriate point on the ais, for eample, =. () From the point in step (), draw line segments parallel to the ais and the ais. () From the ends of the segments in step (), draw additional lines parallel to the ais and the ais to complete the parallelogram. Plane (piece) parallel to the plane: (Fig. 7) () Locate and label the appropriate point on the ais, for eample, =. () From the point in step (), draw line segments parallel to the ais and the ais. () From the ends of the segments in step (), draw additional lines parallel to the ais and the ais to complete the parallelogram. () () in the plane () Fig. 5 Fig. 4 in the plane () () = () () () Fig. 6 () () () () () Fig. 7 () =
Appendi: D Sketches Contemporar Calculus Practice : Sketch the planes = and = in Fig. 8. Practice : Sketch the planes = and = on an XYZ sstem. Practice : Sketch the rectangular bo with opposite corners at (0, 0, 0) and (4,, ). Challenge : Sketch the plane that contains the points (0, 0, ), (4, 0, ), and (0,, ). Fig. 8 C. Ellipses Kes: Plot the vertices and make a parallelogram "frame" for the ellipse. Sketch short tangent segments at the midpoints of the sides of the parallelogram. Eample : An ellipse in the plane with vertices at (,0,0), (,0,0), (0,,0), and (0,,0). (Fig. 9) () Plot the vertices in the plane (=0). () Sketch a parallelogram "frame" b drawing lines through (,0) and (,0) that are parallel to the ais, and lines through (0,) and (0, ) that are parallel to the ais. () At each verte sketch a short "tangent segment" that lies on the side of the parallelogram frame. (4) Finish the sketch b smoothl connecting the "tangent segments." If our sketch requires several conics, it is useful to erase all or most of the parallelogram frame. Steps () & () (4) Fig. 9 Final result
Appendi: D Sketches Contemporar Calculus 4 Eample : An ellipse in the plane with vertices at (,0,0), (,0,0), (0,0,), and (0,0, ). Fig. 0 shows the intermediate steps to construct this ellipse as well as the final result. Practice 4: Sketch the ellipse with vertices at (, 0, 0), (0, 0, 0), (0, 0, ), and (0, 0, ) on the XYZ sstem in Fig.. Practice 5: Sketch the ellipse with vertices at (, 0, ), (, 0, ), (0,, ), and (0,, ). Practice 6: Sketch the ellipse + 9 = on the plane =. Challenge : Sketch the ellipse with vertices at (4,,0), (,0,), (0,,6), and (,4,). D. Parabolas Steps (), () and () Final result Fig. 0 Fig. Kes: Plot the verte and one other point on the parabola. Make a parallelogram "frame" for the parabola and plot a smmetric point. Sketch short "tangent segments" at the plotted points. Eample : Sketch the parabola = 4 + 6 in the plane. (Fig. ) () Plot the verte (,,0). When =, then = () 4() + 6 =. Plot (,,0). () Sketch a parallelogram "frame" b drawing lines through (,,0) parallel to the ais and the ais, and through (,,0) parallel to the ais and the ais. () Plot the "smmetric point" (,,0) and draw line through it parallel to the ais and the ais. (4) Sketch a short "tangent segment" at the verte (,,0) (this "tangent segment" lies on the side of the parallelogram). Add "tangent segments" at the other two plotted points (,,0) and (,,0) (these "tangent segments do are not parallel to the sides of the parallelogram). (5) Finish the sketch b smoothl connecting the "tangent segments."
Appendi: D Sketches Contemporar Calculus 5 Step () () & () (4) (5) Fig. Eample : Sketch the parabola = + in the plane. Fig. shows the intermediate steps to construct this parabola as well as the final result. Practice 7: Sketch the parabola with verte at (0,,0) that contains the point (0,,) on the XYZ sstem in Fig. 4. Practice 8: Sketch the parabola with verte at (,0,) that contains the point (0,0,0). Steps () (4) Fig. Final resul Practice 9: Sketch the parabola = ( ) on the plane =. Challenge : Sketch the parabola with verte (4,,0) that contains the smmetric points (0,,) and (0,,). Fig. 4
Appendi: D Sketches Contemporar Calculus 6 E. General curves in coordinate planes Kes: Start with a regular rectangular coordinate graph of the function and a few "tangent segments." Plot a domain range parallelogram in D along with the points and the "tangent segments." Eample 4: Sketch the curve = sin(), 0, in the plane. (Fig. 5) () Graph = sin(), 0, in the D rectangular = sin( ) coordinate sstem, label a few points and add "tangent segments" at those points. () Sketch the domain range parallelogram, 0 and. Step () () Sketch the points and "tangent segments" in the parallelogram in step (). 0 (4) Finish the sketch b smoothl connecting the "tangent segments." = 0 Practice 0: Sketch the curve = + cos(),, in the plane ( = 0) and the = plane on Step () the XYZ sstem in Fig. 6. Practice : Sketch the curve =, 0 4, in the plane (=0) and the = plane. Practice : Sketch the curve = 4,, in the plane and the = planes () Challenge 4: Sketch the graph of = + on the plane { =, no restrictions on }. (4) Final result Fig. 5 Fig. 6