# How can I plot Trajectory of 3d parametric plot?

x1 and y1 are constants.

I have plotted ParametricPlot3D.

x1= 0.1;
y1= 0.2;
eps = 0.1;
X=ParametricPlot3D[{
(1-ε)*x1+ε*x2,
(1-ε)*y1 + ε*y2,
(1-ε)*x1*y1+ε*x2*y2
}, {x2, 0, 1}, {y2, 0, 1}]


I want to plot trajectory of X when ε is continuously changing 0 to 0.5 as 3D plot. I mean I want to plot 3D space which X can be satisfied when ε is continuously changing 0 to 0.5.

How can I do that?

The below image is producted by adding X each ε. I think we can find better way.

• I think eps should be ε. Anyway, you could try this: x1 = 0.1; y1 = 0.2; Manipulate[ ParametricPlot3D[{(1 - ε)*x1 + ε* x2, (1 - ε)*y1 + ε* y2, (1 - ε)*x1*y1 + ε*x2*y2}, {x2, 0, 1}, {y2, 0, 1}, PlotRange -> {{0, .6}, {0, .6}, {0, .6}}, BoxRatios -> 1] , {ε, 0, 0.5}] – flinty Jul 15 '20 at 12:57

To plot the volume try ParametricRegion as follows:

First do linear interpolation of your expression

f0 = {(1 - \[CurlyEpsilon])*x1 + \[CurlyEpsilon]*x2, (1 - \[CurlyEpsilon])*y1 + \[CurlyEpsilon]*y2, (1 - \[CurlyEpsilon])*x1*y1 + \[CurlyEpsilon]*x2*y2} /. \[CurlyEpsilon] -> 0
f1 = {(1 - \[CurlyEpsilon])*x1 + \[CurlyEpsilon]*x2, (1 - \[CurlyEpsilon])*y1 + \[CurlyEpsilon]*y2, (1 - \[CurlyEpsilon])*x1*y1 + \[CurlyEpsilon]*x2*y2} /. \[CurlyEpsilon] -> 1/2
ip = Function[{x2, y2, u}, Evaluate[ (1/2 - u)/(1/2) f0 + u/(1/2) f1]]


second define and plot a parametric region

reg = ParametricRegion[ip[x2, y2, u], {{u, 0, 1/2}, {x2, 0, 1}, {y2, 0, 1}} ]
Show[Region[reg], MaxRecursion -> 4, Axes -> True]


Perhaps that's the volume you are looking for. How to refine the plot I unfortunately don't know.