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负圆锥体。
现在，我们来构建一个我们称之为“负圆锥体”的东西。要构建一个正圆锥体，我们从平面上剪去一个扇形。在这里，我们添加一个扇形，对应的角度为 q：
(30)
在这张表面上，我们可以用我们的胶带画出测地线，并用三条测地线形成一个三角形。如果您测量角度的总和，您会发现它等于 180° - q。我们将说它定义了一个负曲率的集中区域。
在您的家中也有负曲率的物体。例如，一些座椅： (30 bis)
如果我们取一个圆盘，我们会得到图 (31)：
(31)
当然，如果由三条测地线组成的三角形不包含顶点 S，而顶点 S 包含全部的（负）角曲率，那么总和将是欧几里得几何的总和：180°。
马鞍形。
** **您可以构建大量角度为 - Dq 的基本负圆锥体，并将它们连接起来。您可以通过这种方式使两个相邻顶点之间的距离几乎恒定。这样，您将得到一个 具有恒定负曲率密度的表面：马鞍形。但这个表面永远不会闭合。
一般来说，几何学家称其为 具有恒定负曲率的表面。 (32)
“钝角负圆锥体”。
在前面的部分中，我们使用一个具有恒定正曲率密度的表面（球体的一部分）和一个正圆锥体的一部分，构建了一个钝角正圆锥体。
同样地，我们可以构建一个我们称之为“钝角负圆锥体”的东西。我们需要将马鞍形与负圆锥体的一部分沿它们的公共圆形边缘连接起来。为了确保切平面的连续性，马鞍形中包含的（负）曲率必须等于在构建负圆锥体时所涉及的负曲率。
相对而言，构建所需的负圆锥体部分是相当容易的！ (33)
注意：与正圆锥体一样，负圆锥体也可以用作印刷模具。但很难想象如何将负圆锥体卷到平面上。因此，更容易将平面卷到具有负曲率的模具上。
古腾堡发明了印刷技术。一个浮雕图案被雕刻在一个平面上。然后，将其涂上油墨，并压在平面上。
后来，印刷模具被转换成圆柱形，用于报纸印刷（旋转印刷机）。
但据我所知，没有人使用过圆锥形印刷机。
无论如何，重要的是让两个表面接触，无论采用什么方法。要么移动模具，要么卷动纸张（平面表面）。
如图 (34) 所示，您可以使用圆锥形模具在平面上印刷一些东西。一些圆锥形报纸，平铺。 (34)
没有人会永远使用它，这并不确定。假设您想生产一些具有特殊设计的连衣裙，该设计对应于圆锥对称性。假设您需要生产数千件这样的连衣裙。您可以在圆锥形模具上雕刻图案，然后用它在织物上印刷。顾客可以购买它，并制作“圆锥形”连衣裙，确信所获得的图案在任何地方都是正确的。
图 (35) 展示了使用负曲率模具印刷的结果。右边是一个平铺的负圆锥体。 (35)
图 (36) 展示了如何将马鞍形与负圆锥体的一部分连接起来。
顺便问一下，您可能会问：
- 我如何测量我马鞍形中所包含的负角曲率？
在德克萨斯州的一些地方，靠近数学系，当你买一匹马鞍时，相应的角曲率会标注在附带的票上。如果没有，通过将边缘的周长或面积与根据该负曲率圆盘的半径计算出的欧几里得值进行比较，您可以推断出相应的角曲率。将其视为一个富有成效的练习。 (36)
(37)
现在，我们可以使用我们的胶带，画出测地线并将其投影到平面上，如图 (38) 所示。
(38)
像往常一样，这种平面投影指的是我们的“心灵世界”，即柏拉图洞穴的墙壁。投影的测地线的外观意味着对我们参考物体施加了某种排斥力，例如 排斥性引力。实际上，这一切都应该源自于底层的几何学。
原始版本（英文）
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**Negacones.
**
Let us build now what we will call a "negacone". To build a posicone, we removed a sector from a plane. Here we add one, corresponding to an angle q :
(30)
On this surface we can draw geodesic lines, with our sticky tape and make a triangle with three of them. If you measure the sum of the angles, you will see that it is 180°-q . We will say that it defines a negative curvature concentration.
There are objects with negative curvature in your home. Some seats, for example : (30 bis)
If we take a disk we get the figure (31) :
(31)
Of course, if the triangle composed by the three geodesics does not contain the summit S, which contains all the (negative) angular curvature, the sum will be the euclidean sum : 180°.
The horse saddle.
** **You can build a great number of elementary negacones
with angle - Dq and join them. You can do that in order to have an almost constant distance between two neighbour summits. Then you would build a constant negative curvature density surface : a horse saddle. But this surface will never get closed.
In general the geometer call it a *constant negative curvature surface.
*(32)
"Blunt negacone".
In a precedent section, using a constant positive curvature density surface (a portion of a sphere) and a portion of posicone we built a blunt posicone.
Similarly we can build what we will call a blunt negacone. We have to join a horse saddle to a portion of negacone, along their common circular border. In order to manage the continuity of the tangent plane the (negative) curvature contained in the horse saddle must be equal to the negative curvature involved in the negacone building.
It is relatively easy to build the required portion of a negacone !
(33)
NB : As the posicone, the negacone can be used as a printing matrix. But we can hardly see how to roll a negacone on a flat plane. It is therefore easier to roll the plane on a negative curvature matrix.
Gutemberg invented the printing technique. A design in relief is carved on a plane. Then one puts ink on it and presses it onto a plane.
Later the printing matrix was converted into a cylinder, for newspaper printing (rotative press).
But no one, as far as I know, used the conical press.
In all cases, the important point is to put the two surface into contact, whatever one does. Either you move the matrix, or you roll the paper (the plane surface).
As shown on figure (34) you can use a conical matrix to print something on a plane. Some conical newspaper, put flat.
(34)
It is not definitively sure that nobody will use that, someday. Suppose you want to produce robes, with a special design, corresponding to conical symmetry. Suppose that you have to produce thousands of robes like that. You could engrave the native design on a conical matric, then use it to print on the material,or the fabric. The customer could buy it and make the "conical" robe, being sure that the obtained patter would be good, all over.
On figure (35) is what you get if you print with a negative curvature matrix. On the right a negacone put flat.
(35)
On figure (36) it shows how to join the horse saddle to the portion of negacone.
By the way, you may ask :
- How can I measure the negative angular curvature contained in my horse saddle ?
In some places in Texas, close to mathematical departments, when you buy a horse saddle the corresponding angular curvature is indicated on the joined ticket. If not when you compare the perimeter of the border, or the area,to the euclidean value, calculated from the radius of this regative curvature disk, you can deduce the corresponding angular curvature. Consider this as a fruitfull exercise.
(36)
(37)
Now we can use ou sticky tape, draw geodesics and project it on a plane, as shown on figure (38).
(38)
As usual this plane projection refers to our "mental world", the wall of Plato's cavern. The aspect of the projected geodesic would mean for us that some repulsive force is acting on our reference objects, for example a *repulsive gravitational force *. In fact, all that should come from underlying geometry.