1.
Quadric SurfacesA trace of a surface is the intersection of the surfacewith a plane.
2.
Quadric SurfacesA trace of a surface is the intersection of the surfacewith a plane. The traces of a given surface S that weare interested in are traces parallel to the coordinateplanes,
3.
Quadric SurfacesA trace of a surface is the intersection of the surfacewith a plane. The traces of a given surface S that weare interested in are traces parallel to the coordinateplanes, i.e. the intersection of the surface with theplanes, x = c, y = c, or z = c as shown here.
4.
Quadric SurfacesA trace of a surface is the intersection of the surfacewith a plane. The traces of a given surface S that weare interested in are traces parallel to the coordinateplanes, i.e. the intersection of the surface with theplanes, x = c, y = c, or z = c as shown here. x=c S A trace with x = c
5.
Quadric SurfacesA trace of a surface is the intersection of the surfacewith a plane. The traces of a given surface S that weare interested in are traces parallel to the coordinateplanes, i.e. the intersection of the surface with theplanes, x = c, y = c, or z = c as shown here. x=c y=c S S A trace with x = c A trace with y = c
6.
Quadric SurfacesA trace of a surface is the intersection of the surfacewith a plane. The traces of a given surface S that weare interested in are traces parallel to the coordinateplanes, i.e. the intersection of the surface with theplanes, x = c, y = c, or z = c as shown here. z=c x=c y=c S S S A trace with x = c A trace with y = c A trace with z = c
7.
Quadric SurfacesA trace of a surface is the intersection of the surfacewith a plane. The traces of a given surface S that weare interested in are traces parallel to the coordinateplanes, i.e. the intersection of the surface with theplanes, x = c, y = c, or z = c as shown here. z=c x=c y=c S S S A trace with x = c A trace with y = c A trace with z = cBecause hand point–plotting is not useful in R3, onemethod to visualize a surface is the trace-method.
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Quadric SurfacesParallel traces are similar to thesuccessive ribs of a structure thatshow the contour of its surface.
9.
Quadric SurfacesParallel traces are similar to thesuccessive ribs of a structure thatshow the contour of its surface.With the trace–method, usingparallel planes, we construct aseries of traces on the surface Sto reveal the shape of the S.
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Quadric SurfacesParallel traces are similar to thesuccessive ribs of a structure thatshow the contour of its surface.With the trace–method, usingparallel planes, we construct aseries of traces on the surface Sto reveal the shape of the S.The planes x = c, y = c, or z = cproduce traces that areperpendicular to the x, y, and zaxes respectively.
11.
Quadric SurfacesParallel traces are similar to thesuccessive ribs of a structure thatshow the contour of its surface.With the trace–method, usingparallel planes, we construct aseries of traces on the surface Sto reveal the shape of the S. Traces with x = c.The planes x = c, y = c, or z = cproduce traces that areperpendicular to the x, y, and zaxes respectively. For example,the traces of S with the x = c,as c varies, are plane curves thatstacked perpendicularly to the x–axis.
12.
Quadric SurfacesParallel traces are similar to thesuccessive ribs of a structure thatshow the contour of its surface.With the trace–method, usingparallel planes, we construct aseries of traces on the surface Sto reveal the shape of the S. Traces with x = c.The planes x = c, y = c, or z = cproduce traces that areperpendicular to the x, y, and z xaxes respectively. For example,the traces of S with the x = c,as c varies, are plane curves thatstacked perpendicularly to the x–axis.
13.
Quadric SurfacesParallel traces are similar to thesuccessive ribs of a structure thatshow the contour of its surface.With the trace–method, usingparallel planes, we construct aseries of traces on the surface Sto reveal the shape of the S. Traces with x = c.The planes x = c, y = c, or z = cproduce traces that areperpendicular to the x, y, and z xaxes respectively. For example,the traces of S with the x = c,as c varies, are plane curves thatstacked perpendicularly to the x–axis.
14.
Quadric SurfacesParallel traces are similar to thesuccessive ribs of a structure thatshow the contour of its surface.With the trace–method, usingparallel planes, we construct aseries of traces on the surface Sto reveal the shape of the S. Traces with x = c.The planes x = c, y = c, or z = cproduce traces that areperpendicular to the x, y, and z xaxes respectively. For example,the traces of S with the x = c,as c varies, are plane curves thatstacked perpendicularly to the x–axis.
15.
Quadric SurfacesParallel traces are similar to thesuccessive ribs of a structure thatshow the contour of its surface.With the trace–method, usingparallel planes, we construct aseries of traces on the surface Sto reveal the shape of the S. Traces with x = c.The planes x = c, y = c, or z = cproduce traces that areperpendicular to the x, y, and z xaxes respectively. For example,the traces of S with the x = c, Sas c varies, are plane curves thatstacked perpendicularly to the x–axis.
16.
Quadric SurfacesWe will examine quadratic surfaces, i.e. the graphs of2nd degree equations of the formAx2 + By2 + Cz2 + Dx + Ey + Fz + G = 0. We will studythe ones that can be transformed to one of the followingforms (with x2 as the leading term): x2 y2 z2 x2 y2 + + =1 z = 2 + 2 a2 b2 c 2 a2 b All positive square terms Constant term = 0 x2 y2 z2 + – =1 x2 y2 a 2 b 2 c 2 z = + Assuming no z2One negative square term a2 b2 x2 y2 z2 x2 – – =1 y2 Assuming no z2 a2 b2 c 2 z = – a2 b2Two negative square termsTo sketch then surfaces, if possible, find the square-sums who give traces that are ellipses or circles.
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Quadric SurfacesTo graph the trace of f(x, y, z) = 0 with y = c,for example, set y = c to obtain f(x, c, z) = 0,an xz-equation.
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Quadric SurfacesTo graph the trace of f(x, y, z) = 0 with y = c,for example, set y = c to obtain f(x, c, z) = 0,an xz-equation. The trace is the 2D graph off(x, c, z) = 0 in the plane y = c.
19.
Quadric SurfacesTo graph the trace of f(x, y, z) = 0 with y = c,for example, set y = c to obtain f(x, c, z) = 0,an xz-equation. The trace is the 2D graph off(x, c, z) = 0 in the plane y = c. The traces of quadricsurfaces, after setting a variable to constants, areconic (2nd degree) sections in the planes.
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Quadric SurfacesTo graph the trace of f(x, y, z) = 0 with y = c,for example, set y = c to obtain f(x, c, z) = 0,an xz-equation. The trace is the 2D graph off(x, c, z) = 0 in the plane y = c. The traces of quadricsurfaces, after setting a variable to constants, areconic (2nd degree) sections in the planes. Circles: Ellipses: b x2+y2=r2 r x2 + y2 a a2 =1 b2 Hyperbolas: b Parabolas: x2 – y 2 a a2 b2 = 1 y = ax2 Figures are shown centered at (0, 0), in gereral they centered at (h, k).
22.
Quadric Surfaces x2 + y2 + z2 = 1The graphs of a2 b2 c2Isolate the sum of two square terms, e.g. x2 and y2: x2 y2 = 1 – z2 a2 + b2 c2
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Quadric SurfacesThe graphs of a2 x2 + y2 + z2 = 1 b2 c2Isolate the sum of two square terms, e.g. x2 and y2: x2 y2 = 1 – z2 a2 + b2 c2If z > |c|, we getx2 + y2 = negative #,a2 b2so there is no graph in thez = c plane.
24.
Quadric SurfacesThe graphs of a2 x2 + y2 + z2 = 1 b2 c2Isolate the sum of two square terms, e.g. x2 and y2: x2 y2 = 1 – z2 a2 + b2 c2If z > |c|, we getx2 + y2 = negative #,a2 b2so there is no graph in thez = c plane.If z = |c|, we get 2x2 + y2 we get the two poles. 2 = a b 0,
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Quadric SurfacesThe graphs of a2 x2 + y2 + z2 = 1 b2 c2Isolate the sum of two square terms, e.g. x2 and y2: x2 y2 = 1 – z2 z a2 + b2 c2 cIf z > |c|, we get bx + y = negative #, 2 2 a ya2 b2 x z=0so there is no graph inthez = c plane. get x2If z = |c|, we y2 a 2 + b 2 = we get the two poles. y2 0,If z < |c|, we get x 2 + b2 = positive #, 2 awhich are ellipses, with the trace z = 0 being thelargest elliptic trace forming the equator.
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Quadric Surfaces x2 y2 z2 = 1The graphs of + – 2 a 2 b 2 c
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Quadric Surfaces x2 y2 z2 = 1The graphs of + – 2 a 2 b 2 cExample A. Graph x2 + y2 – z2 = 1
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Quadric Surfaces x2 y2 z2 = 1The graphs of + – 2 a 2 b 2 cExample A. Graph x2 + y2 – z2 = 1Solve for the sum–of–two–squares: x2 + y2 = 1 + z2
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Quadric Surfaces x2 y2 z2 = 1The graphs of + – 2 a 2 b 2 cExample A. Graph x2 + y2 – z2 = 1Solve for the sum–of–two–squares: x2 + y2 = 1 + z2Let z varies, we getcircles in the z-planes.
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Quadric Surfaces x2 y2 z2 = 1The graphs of + – 2 a 2 b 2 cExample A. Graph x2 + y2 – z2 = 1Solve for the sum–of–two–squares: x2 + y2 = 1 + z2Let z varies, we getcircles in the z-planes.When z = 0, we get thesmallest circle
31.
Quadric Surfaces x2 y2 z2 = 1The graphs of + – 2 a 2 b 2 cExample A. Graph x2 + y2 – z2 = 1Solve for the sum–of–two–squares: x2 + y2 = 1 + z2Let z varies, we getcircles in the z-planes.When z = 0, we get thesmallest circle and as|z| gets larger, the circlesget larger.
32.
Quadric Surfaces x2 y2 z2 = 1The graphs of + – 2 a 2 b 2 cExample A. Graph x2 + y2 – z2 = 1Solve for the sum–of–two–squares: x2 + y2 = 1 + z2Let z varies, we getcircles in the z-planes.When z = 0, we get thesmallest circle and as|z| gets larger, the circlesget larger. These circlesare stacked vertically.
33.
Quadric Surfaces x2 y2 z2 = 1The graphs of + – 2 a 2 b 2 cExample A. Graph x2 + y2 – z2 = 1Solve for the sum–of–two–squares: x2 + y2 = 1 + z2Let z varies, we getcircles in the z-planes.When z = 0, we get thesmallest circle and as|z| gets larger, the circles y –z 2 2 =1get larger. These circlesare stacked vertically.Set x = 0, we get thehyperbolic outline of the"stacked circles“.
34.
Quadric Surfaces x2 y2 z2 = 1The graphs of + – 2 a 2 b 2 cExample A. Graph x2 + y2 – z2 = 1Solve for the sum–of–two–squares: x2 + y2 = 1 + z2Let z varies, we getcircles in the z-planes.When z = 0, we get thesmallest circle and as|z| gets larger, the circles y –z =1 2 2get larger. These circlesare stacked vertically.Set x = 0, we get thehyperbolic outline of the"stacked circles“.
35.
Quadric Surfaces x2 y2 z2Summary for the graphs of a2 + b2 – c2 = 1
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Quadric Surfaces x2 y2 z2Summary for the graphs of a2 + b2 – c2 = 1 x2 y2 z2i. Solve for the sum of squares: a2 + b2 = 1 + c2
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Quadric Surfaces x2 y2 z2Summary for the graphs of a2 + b2 – c2 = 1 x2 y2 z2i. Solve for the sum of squares: a2 + b2 = 1 + c2ii. Let z varies,for a fixed k, we getan ellipses in the z = k planes,
38.
Quadric Surfaces x2 y2 z2Summary for the graphs of a2 + b2 – c2 = 1 x2 y2 z2i. Solve for the sum of squares: a2 + b2 = 1 + c2ii. Let z varies,for a fixed k, we getan ellipses in the z = k planes,with z = 0 giving the smallestellipse x2/a2+y2/b2 =1 as the waistand progressively larger ellipsesstacked vertically.
39.
Quadric Surfaces x2 y2 z2Summary for the graphs of a2 + b2 – c2 = 1 x2 y2 z2i. Solve for the sum of squares: a2 + b2 = 1 + c2ii. Let z varies, z traces: x 2 2 y k2for a fixed k, we get a 2 + b = 1+ 2 c2an ellipses in the z = k planes, z = kwith z = 0 giving the smallestellipse x2/a2+y2/b2 =1 as the waistand progressively larger ellipsesstacked vertically. x2 y2 – z2 = 1 a2 + b2 c2
40.
Quadric Surfaces x2 y2 z2Summary for the graphs of a2 + b2 – c2 = 1 x2 y2 z2i. Solve for the sum of squares: a2 + b2 = 1 + c2ii. Let z varies, z traces: x 2 2 y k2for a fixed k, we get a 2 + b = 1+ 2 c2an ellipses in the z = k planes, z = kwith z = 0 giving the smallestellipse x2/a2+y2/b2 =1 as the waistand progressively larger ellipsesstacked vertically.Setting x = 0 or y =0, we have thehyperbolic outlines of the surface. x2 y2 – z2 = 1 a2 + b2 c2
41.
Quadric Surfaces x2 y2 z2Summary for the graphs of a2 + b2 – c2 = 1 x2 y2 z2i. Solve for the sum of squares: a2 + b2 = 1 + c2ii. Let z varies, z traces: x 2 2 y k 2for a fixed k, we get a 2 + b = 1+ c 2 2an ellipses in the z = k planes, z = kwith z = 0 giving the smallestellipse x2/a2+y2/b2 =1 as the waistand progressively larger ellipsesstacked vertically.Setting x = 0 or y =0, we have thehyperbolic outlines of the surface.Exercise: Draw a few traces along the x2 y2 – z2 a2 + b2 c2 = 1x and y axes and find their equations.
43.
Quadric SurfacesIf we fix the a, b and c in x2 y2 z2 + 2 = 1+ 2 a2 b cbut we replace 1 by k so we have x2 + y2 z2 = k + c2 a2 b2 x2 + y2 = 1 + z2
44.
Quadric SurfacesIf we fix the a, b and c in x2 y2 z2 + 2 = 1+ 2 a2 b cbut we replace 1 by k so we have x2 + y2 z2 = k + c2 a2 b2then as k → 0 the waists of thecorresponding surfaces shrink, x2 + y2 = 1 + z2
45.
Quadric SurfacesIf we fix the a, b and c in x2 y2 z2 + 2 = 1+ 2 a2 b cbut we replace 1 by k so we have x2 + y2 z2 = k + c2 a2 b2then as k → 0 the waists of thecorresponding surfaces shrink, x 2 + y2 = 1 + z2and if k = 0 we have the equationz2 = x2 y2 + 2c 2 a2 bwhose waist is just one point as inthe case of z2 = x2 + y2 shown here. x2 + y2 = 0 + z2
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Quadric SurfacesIf we fix the a, b and c in x2 y2 z2 + 2 = 1+ 2 a2 b cbut we replace 1 by k so we have x2 + y2 z2 = k + c2 a2 b2then as k → 0 the waists of thecorresponding surfaces shrink, x 2 + y2 = 1 + z2and if k = 0 we have the equationz2 = x2 y2 + 2c 2 a2 b z = ±ywhose waist is just one point as inthe case of z2 = x2 + y2 shown here.The x = 0 traces give the outlinez = ±y in the yz–plane. x +y 2 2 = 0 + z2
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Quadric Surfaces z2 x2 y2The graphs of – – 2 = 1 a2 b2 c
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Quadric Surfaces z2 x2 y2The graphs of a2 – b2 – c2 = 1 x2 y2 = z2 – 1Solve for the sum of squares: 2 + 2 a b c2
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Quadric Surfaces z2 x2 y2The graphs of a2 – b2 – c2 = 1 x2 y2 = z2 – 1Solve for the sum of squares: 2 + 2 a b c2 x2 + y2 z2it’s equation if we set k = –1 in 2 = k + 2 a2 b c
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Quadric Surfaces z2 x2 y2The graphs of a2 – b2 – c2 = 1 x2 y2 = z2 – 1Solve for the sum of squares: 2 + 2 a b c2 x2 + y2 z2it’s equation if we set k = –1 in 2 = k + 2 a2 b cIn this case, we havenon–empty level curves if z2 – 1 ≥ 0, i.e. |z| ≥ c. c2
51.
Quadric Surfaces z2 x2 y2The graphs of a2 – b2 – c2 = 1 x2 y2 = z2 – 1Solve for the sum of squares: 2 + 2 a b c2 x2 + y2 z2it’s equation if we set k = –1 in 2 = k + 2 a2 b cIn this case, we havenon–empty level curves if z2 – 1 ≥ 0, i.e. |z| ≥ c. c2For |z| > c, we get large andlarger ellipses .
52.
Quadric Surfaces z2 x2 y2The graphs of a2 – b2 – c2 = 1 x2 y2 = z2 – 1Solve for the sum of squares: 2 + 2 a b c2 x2 + y2 z2it’s equation if we set k = –1 in 2 = k + 2 a2 b cIn this case, we havenon–empty level curves if z2 – 1 ≥ 0, i.e. |z| ≥ c. c2For |z| > c, we get large andlarger ellipses .The surfacex2 + y2 = z2 – 1is shown here with x +y 2 2 = z2 – 1some x and z traces.
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Quadric SurfacesSummary of The graphs of z2 = x2 + y2 + k: z2 = x2 + y2 + 1 z2 = x2 + y2 + 0 z2 = x2 + y2 – 1 x2 y2 z2 x2 y2 + + =1 z = 2 + 2 a 2 b2 c 2 a2 b All positive square terms Constant term = 0 x2 y2 z2 Hence we have addressed + – =1 a2 b2 c2 One negative square term cover these cases. x2 y2 z2 The following is a summary – – =1 a2 b2 c 2 and the remaining cases. Two negative square terms
54.
Quadric Surfaces x2 y2 z2 x2 y2 z2 + + =1 + – =0 a2 b2 c 2 a2 b 2 c 2All positive square terms Constant term = 0x2 y2 z2 z2 x2 y2 + – =1 – – =1a2 b2 c 2 a2 b 2 c 2One negative square term Two negative square termsThe remaining cases have a missing square term.We will assume that there is no z2. We may in thesecases, express z as a function of x and y.Again, there are two cases; the sum or the differenceof squares. x2 – y2 z = x2 + y2 2 2 z= 2 a b a b2 Assuming no z2 Assuming no z2
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Quadric SurfacesTraces with z = c are called level curves since zcoordinates give the height of the poitnt.
56.
Quadric SurfacesTraces with z = c are called level curves since zcoordinates give the height of the poitnt.Example A. a. Sketch the traces of z – x2 – y2 = 0with z = –1, 0, 1, 4.
57.
Quadric SurfacesTraces with z = c are called level curves since zcoordinates give the height of the poitnt.Example A. a. Sketch the traces of z – x2 – y2 = 0with z = –1, 0, 1, 4.Set z = x2 + y2, let z = –1, 0, 1, 4 and we get thefollowing xy-equations:z = –1: –1 = x2 + y2z = 0: 0 = x2 + y2z = 1: 1 = x2 + y2z = 4: 4 = x2 + y2
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Quadric SurfacesTraces with z = c are called level curves since zcoordinates give the height of the poitnt.Example A. a. Sketch the traces of z – x2 – y2 = 0with z = –1, 0, 1, 4.Set z = x2 + y2, let z = –1, 0, 1, 4 and we get thefollowing xy-equations:z = –1: –1 = x2 + y2 (no solution)z = 0: 0 = x2 + y2z = 1: 1 = x2 + y2z = 4: 4 = x2 + y2
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Quadric SurfacesTraces with z = c are called level curves since zcoordinates give the height of the poitnt.Example A. a. Sketch the traces of z – x2 – y2 = 0with z = –1, 0, 1, 4.Set z = x2 + y2, let z = –1, 0, 1, 4 and we get thefollowing xy-equations:z = –1: –1 = x2 + y2 (no solution)z = 0: 0 = x2 + y2z = 1: 1 = x2 + y2z = 4: 4 = x2 + y2 z=0 y x
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Quadric SurfacesTraces with z = c are called level curves since zcoordinates give the height of the poitnt.Example A. a. Sketch the traces of z – x2 – y2 = 0with z = –1, 0, 1, 4.Set z = x2 + y2, let z = –1, 0, 1, 4 and we get thefollowing xy-equations:z = –1: –1 = x2 + y2 (no solution) z=4z = 0: 0 = x2 + y2 z=1z = 1: 1 = x2 + y2z = 4: 4 = x2 + y2 z=0 y x
61.
Quadric SurfacesTraces with z = c are called level curves since zcoordinates give the height of the poitnt.Example A. a. Sketch the traces of z – x2 – y2 = 0with z = –1, 0, 1, 4.Set z = x2 + y2, let z = –1, 0, 1, 4 and we get thefollowing xy-equations:z = –1: –1 = x2 + y2 (no solution) z=4z = 0: 0 = x2 + y2 z=1z = 1: 1 = x2 + y2z = 4: 4 = x2 + y2 z=0We note that for z < 0, there is yno trace and as z →∞, we havesuccessive higher and larger xcircles.
62.
Quadric SurfacesNext we reconstruct the traces along the x–axis forthe outline of the vertically stacked circles.
63.
Quadric SurfacesNext we reconstruct the traces along the x–axis forthe outline of the vertically stacked circles.b. Sketch the traces of z – x2 – y2 = 0 withx = –2, –1, 0, 1, 2.
64.
Quadric Surfaces Next we reconstruct the traces along the x–axis for the outline of the vertically stacked circles. b. Sketch the traces of z – x2 – y2 = 0 with x = –2, –1, 0, 1, 2.Set z = x2 + y2, let x = –2, –1, 0, 1, 2 and get thefollowing yz-equations:
65.
Quadric Surfaces Next we reconstruct the traces along the x–axis for the outline of the vertically stacked circles. b. Sketch the traces of z – x2 – y2 = 0 with x = –2, –1, 0, 1, 2.Set z = x2 + y2, let x = –2, –1, 0, 1, 2 and get thefollowing yz-equations:x = –2: z = 4 + y2x = –1: z = 1 + y2x = 0: z = y2x = 1: z = 1 + y2x = 2: z = 4 + y2
66.
Quadric Surfaces Next we reconstruct the traces along the x–axis for the outline of the vertically stacked circles. b. Sketch the traces of z – x2 – y2 = 0 with x = –2, –1, 0, 1, 2.Set z = x2 + y2, let x = –2, –1, 0, 1, 2 and get thefollowing yz-equations:x = –2: z = 4 + y2x = –1: z = 1 + y2x = 0: z = y2x = 1: z = 1 + y2x = 2: z = 4 + y2 x=0 x
67.
Quadric Surfaces Next we reconstruct the traces along the x–axis for the outline of the vertically stacked circles. b. Sketch the traces of z – x2 – y2 = 0 with x = –2, –1, 0, 1, 2.Set z = x2 + y2, let x = –2, –1, 0, 1, 2 and get thefollowing yz-equations:x = –2: z = 4 + y2x = –1: z = 1 + y2x = 0: z = y2x = 1: z = 1 + y2x = 2: z = 4 + y2 x=0 x=1 x
68.
Quadric Surfaces Next we reconstruct the traces along the x–axis for the outline of the vertically stacked circles. b. Sketch the traces of z – x2 – y2 = 0 with x = –2, –1, 0, 1, 2.Set z = x2 + y2, let x = –2, –1, 0, 1, 2 and get thefollowing yz-equations:x = –2: z = 4 + y2x = –1: z = 1 + y2x = 0: z = y2x = 1: z = 1 + y2x = 2: z = 4 + y2 x=0 x=1 x x=2
69.
Quadric Surfaces Next we reconstruct the traces along the x–axis for the outline of the vertically stacked circles. b. Sketch the traces of z – x2 – y2 = 0 with x = –2, –1, 0, 1, 2.Set z = x2 + y2, let x = –2, –1, 0, 1, 2 and get thefollowing yz-equations:x = –2: z = 4 + y2x = –1: z = 1 + y2x = 0: z = y2x = 1: z = 1 + y2 x = –2x = 2: z = 4 + y2 x = –1 x=0 x=1 x x=2
70.
Quadric Surfaces Next we reconstruct the traces along the x–axis for the outline of the vertically stacked circles. b. Sketch the traces of z – x2 – y2 = 0 with x = –2, –1, 0, 1, 2.Set z = x2 + y2, let x = –2, –1, 0, 1, 2 and get thefollowing yz-equations:x = –2: z = 4 + y2x = –1: z = 1 + y2x = 0: z = y2x = 1: z = 1 + y2 x = –2x = 2: z = 4 + y2 x = –1We put parts a and b together x=0to visualize the surface. x=1 x x=2
71.
Quadric SurfacesThe graph of of z – x2 – y2 = 0 or z = x2 + y2
72.
Quadric SurfacesThe graph of of z – x2 – y2 = 0 or z = x2 + y2
73.
Quadric SurfacesThe graph of of z – x2 – y2 = 0 or z = x2 + y2
74.
Quadric SurfacesThe graph of of z – x2 – y2 = 0 or z = x2 + y2
75.
Quadric SurfacesThe graph of of z – x2 – y2 = 0 or z = x2 + y2
76.
Quadric Surfaces x2 y2 z2 x2 y2 z2 + + =1 + – =0 a2 b2 c 2 a2 b c 2 2All positive square terms Constant term = 0 x2 y2 z2 z2 x2 y2 + – =1 – – =1 a2 b2 c 2 a2 b 2 c 2One negative square term Two negative square terms x2 y2 x2 y2z = + z = – a 2 b 2 a 2 b2Assuming no z2 Assuming no z2
77.
Quadric Surfaces x2 y2 z2 x2 y2 z2 + + =1 + – =0 a2 b2 c 2 a2 b c 2 2All positive square terms Constant term = 0 x2 y2 z2 z2 x2 y2 + – =1 – – =1 a2 b2 c 2 a2 b 2 c 2One negative square term Two negative square terms x2 y2 x2 y2z = + z = – a 2 b 2 a 2 b2Assuming no z2 Assuming no z2The graph of z = x2 – y2
78.
Quadric Surfaces x2 y2 z2 x2 y2 z2 + + =1 + – =0 a2 b2 c 2 a2 b c 2 2 All positive square terms Constant term = 0 x2 y2 z2 z2 x2 y2 + – =1 – – =1 a2 b2 c 2 a2 b 2 c 2 One negative square term Two negative square terms x2 y2 x2 y2z = + z = – a 2 b 2 a 2 b2Assuming no z2 Assuming no z2The graph of z = x2 – y2There is no ellipse to track.
79.
Quadric Surfaces x2 y2 z2 x2 y2 z2 + + =1 + – =0 a2 b2 c 2 a2 b c 2 2All positive square terms Constant term = 0 x2 y2 z2 z2 x2 y2 + – =1 – – =1 a2 b2 c 2 a2 b 2 c 2One negative square term Two negative square terms x2 y2 x2 y2z = + z = – a 2 b 2 a 2 b2Assuming no z2 Assuming no z2The graph of z = x2 – y2There is no ellipse to track.We may track parabolas byletting x = c, we get z = c2 – y2which are parabolas withmaximum at (0, c) in the x planes.
80.
Quadric Surfaces x2 y2 z2 x2 y2 z2 + + =1 + – =0 a2 b2 c 2 a2 b c 2 2All positive square terms Constant term = 0 x2 y2 z2 z2 x2 y2 + – =1 – – =1 a2 b2 c 2 a2 b 2 c 2One negative square term Two negative square terms x2 y2 x2 y2z = + z = – a 2 b 2 a 2 b2Assuming no z2 Assuming no z2The graph of z = x2 – y2 Traces with x = c.There is no ellipse to track.We may track parabolas by xletting x = c, we get z = c – y 2 2which are parabolas withmaximum at (0, c) in the x planes.
81.
Quadric Surfaces x2 y2 z2 x2 y2 z2 + + =1 + – =0 a2 b2 c 2 a2 b c 2 2All positive square terms Constant term = 0 x2 y2 z2 z2 x2 y2 + – =1 – – =1 a2 b2 c 2 a2 b 2 c 2One negative square term Two negative square terms x2 y2 x2 y2z = + z = – a 2 b 2 a 2 b2Assuming no z2 Assuming no z2The graph of z = x2 – y2 Traces with x = c.There is no ellipse to track.We may track parabolas by xletting x = c, we get z = c – y 2 2which are parabolas withmaximum at (0, c) in the x planes.
82.
Quadric Surfaces x2 y2 z2 x2 y2 z2 + + =1 + – =0 a2 b2 c 2 a2 b c 2 2All positive square terms Constant term = 0 x2 y2 z2 z2 x2 y2 + – =1 – – =1 a2 b2 c 2 a2 b 2 c 2One negative square term Two negative square terms x2 y2 x2 y2z = + z = – a 2 b 2 a 2 b2Assuming no z2 Assuming no z2The graph of z = x2 – y2 Traces with x = c.There is no ellipse to track.We may track parabolas by xletting x = c, we get z = c – y 2 2which are parabolas withmaximum at (0, c) in the x planes.
83.
Quadric SurfacesIf c = 0, the vertex is at z = 0. xTraces with x = c.
84.
Quadric SurfacesIf c = 0, the vertex is at z = 0. The larger the |x| is,the higher the vertex z = c thus the parabola is.These parabolas are stacked in the x-direction. xTraces with x = c.
85.
Quadric SurfacesIf c = 0, the vertex is at z = 0. The larger the |x| is,the higher the vertex z = c thus the parabola is.These parabolas are stacked in the x-direction. xTraces with x = c.
86.
Quadric SurfacesIf c = 0, the vertex is at z = 0. The larger the |x| is,the higher the vertex z = c thus the parabola is.These parabolas are stacked in the x-direction.We obtain the saddle with the origin as a saddle point. xTraces with x = c.
87.
Quadric SurfacesIf c = 0, the vertex is at z = 0. The larger the |x| is,the higher the vertex z = c thus the parabola is.These parabolas are stacked in the x-direction.We obtain the saddle with the origin as a saddle point.The saddle point is the pass between two hills. xTraces with x = c. The surface z = x2 – y2 with (0, 0, 0) as the saddle point
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