# how can I bold the outline of conic section as the plane intercepts the cone?

Plot3D[{-5 - x - y, -Sqrt[8 x^2 + 8 y^2]}, {x, -5, 5}, {y, -5, 5},
Mesh -> None, BoxRatios -> {1, 1, 1}, PlotLegends -> "Expressions"]


• You could use Solve to find the equation of the intersection and then use ParametricPlot3D to plot it. Combine it with the other plot using Show. Mar 18, 2015 at 18:51
• Alternatively, you could use RegionIntersection with Cone and InfinitePlane. Mar 18, 2015 at 18:56
• possible duplicate of About Slicing through Graphics3D
– Jens
Mar 18, 2015 at 19:13
• @jens It is my impression (but I may be wrong here) that the OP wants to see the plane and cone and their intersection (accentuated). The post you referred to has the plane slice off a piece of the cone, which is not the same. Mar 18, 2015 at 20:55
• To the closers: The OP has update the question to indicate that she wants the outline of the intersection be marked. This is not the case in the question linked as duplicate. Mar 18, 2015 at 22:38

Using BoundaryStyle

Use the option BoundaryStyle and set the option value to {{1, 2} -> Directive[Thick, Red]}:

Plot3D[{-5 - x - y, -Sqrt[8 x^2 + 8 y^2]}, {x, -5, 5}, {y, -5, 5},
Mesh -> None, BoxRatios -> {1, 1, 1},
BoundaryStyle -> {{1, 2} -> Directive[Thick, Red]} ]


Note: This particular usage for BoundaryStyle in not documented. The earliest reference on this site is this answer by Daniel Lichtblau

Using MeshFunctions

Use the difference between the two functions as the setting for option MeshFunctions:

Plot3D[{-5 - x - y, -Sqrt[8 x^2 + 8 y^2]}, {x, -5, 5}, {y, -5, 5},
MeshFunctions -> {-5 - # - #2 - (-Sqrt[8 #^2 + 8 #2^2]) &},
Mesh -> {{{0, Directive[Red, Thick]}}}, BoxRatios -> {1, 1, 1},
PlotLegends -> "Expressions"]


• +1 Of course. Interestingly, your specific application of BoundaryStyle does not seem to be documented on its documentation page. Where did you learn about this? Mar 19, 2015 at 13:28
• @SjoerdC.deVries, thanks for the vote. It is not in documentation. I have a faint memory of having seen something like this on this site; but i could not find it with the usual key words. Btw, it does not work in some cases where one would wish it would, e.g. i could not find a way to make it work in [this q/a[(mathematica.stackexchange.com/q/58045/125)
– kglr
Mar 19, 2015 at 13:45
• @SjoerdC.deVries, found the reference: Daniel's answer here
– kglr
Apr 14, 2015 at 20:53
• Ah, that's great! I see that I voted for it, so I should have known. Apr 14, 2015 at 22:09

Find equations for intersection:

Solve[-5 - x - y == -Sqrt[8 x^2 + 8 y^2], x]


{{x -> 1/7 (5 + y - 2 Sqrt[2] Sqrt[25 + 10 y - 6 y^2])}, {x -> 1/7 (5 + y + 2 Sqrt[2] Sqrt[25 + 10 y - 6 y^2])}}

Range of y:

Solve[25 + 10 y - 6 y^2 == 0, y]


{{y -> 5/6 (1 - Sqrt[7])}, {y -> 5/6 (1 + Sqrt[7])}}

Draw intersection:

inter =
With[
{
x1 = 1/7 (5 + y - 2 Sqrt[2] Sqrt[25 + 10 y - 6 y^2]),
x2 = 1/7 (5 + y + 2 Sqrt[2] Sqrt[25 + 10 y - 6 y^2])
},
ParametricPlot3D[{{x1, y, -5 - x1 - y}, {x2, y, -5 - x2 - y}},
{y, 5/6 (1 - Sqrt[7]), 5/6 (1 + Sqrt[7])},
PlotStyle -> Blue
]
]


cone = Cone[{{0, 0, -Sqrt[200]}, {0, 0, 0}}, 5];