Intersection of surfaces
[list][*]two types: partial intersection v. complete interpenetration[/*][*]the intersection will be a spatial polyline[/*][*]in case of partial intersection -> one polyline[/*][*]in case of complete interpenetration -> two polylines[/*][*]construction method: with the help of "simple" intersections[/*][*]see: [url=https://www.geogebra.org/m/kc5ec9df#material/g7zg6ggn]auxiliary figure[/url] | [url=https://www.geogebra.org/m/fxjypmny#material/a2ksvfan]3D figure[/url][/*][/list]
We will study the easier cases of intersections of...[br][list][*]cylinders,[/*][*]cones,[/*][*]spheres,[/*][*]toruses.[/*][/list][br][url=https://www.geogebra.org/m/hewqqytj]Interactive playground[/url] (as well as help for the drawing assignment) [sub](by Ms. Johanna Pék Ph.D.)[/sub]
[list][/list][size=100][list] [*][b]Partial intersection[/b]: if the two surfaces intersect each other so that the intersection curve has only [u]one branch[/u][/*] [*][b]Intersection with double points[/b]: if the [u]two surfaces are tangent[/u] to each other and the curve will have one or two double points, id est it will intersect itself; in some special cases the so called 'decomposition' occurs (the intersection curve desintegrates into two - or sometimes more - pieces). [url=https://www.geogebra.org/m/netyguuu][u]Interactive figure [/u][/url]of double point intersections in case of a cylinder and a cone by Ms. Johanna Pék Ph.D.[/*] [*][b]Complete interpenetration[/b]:[br] if the two surfaces intersect each other so that one of them is divided into two separate parts by the other one and the intersection curve has [u]two[/u] (or sometimes more) [u]branches[/u].[/*][/list][/size]
[list][*][b]General case[/b]: the degree of the intersection curve is the [u]product of the degree of the surfaces[/u]. We have studied three quadratic surfaces (circular cylinder and cone, sphere) and a quartic one (torus). In case of the intersection of [u]quadratic surfaces[/u] the result will be a [u]quartic space curve[/u] (2x2=4). In case of the intersection of a quadratic surface and a torus the result will be a space curve of 8th degree. A nice example of a quartic curve is the so called Viviani-curve (see [url=https://www.geogebra.org/m/t8tensp6][u]interactive figure[/u][/url] by Ms. Johanna Pék Ph.D.), which is a one-double-point intersection of a sphere and a cylinder, if the radius of the cylinder is half of that of the sphere.[/*][/list]
[list][/list][i]The intersection of a sphere and a cylinder is usually a quartic curve of one or two branches.[/i]
[i]An example on a quartic space curve as the intersection curve of a cone and a cylinder (two quadratic surfaces: 2x2=4). [url=https://www.geogebra.org/m/rwqufqsv]Interactive figure[/url][/i]
[list][*][b]Special cases[/b]: in case of surfaces of revolution, the intersection curve can have a lower degree than expected if they have parallel (or coinciding) axes of revolution. In case of the intersection of [u]two spheres,[/u] the result will be a [u]single circe[/u] (quadratic curve). In case of [u]circular cylinders having parallel generators[/u], the result will be [u]two parallel straight lines[/u] (the common generators). In case of the intersection of a [u]sphere and a torus[/u], the intersection will be a [u]quartic space curve[/u]. In case of [u]surfaces of revolution sharing their axes of revolution[/u], the intersection curve(s) will be one or more [u]parallel circles[/u]. Note: the advantage of these simplified curves (id est the common parallel circles of a sphere and other surfaces of revolution) is that we can use them to construct points of intersection.[/*][/list]
[list][/list][i][size=100]If the axis of the cylinder passes through the center of the sphere, the intersection curve consists of two circles.[/size][/i]
[list][/list][i]Sphere and torus might intersect each other along common parallel circles[/i]
[i]Two spheres intersect each other along circles (think of bubbles). [url=https://www.geogebra.org/m/yvcx7s8h]Interactive figure[/url][/i]
[list][*][b]Degenerated intersection curves[/b]: under special circumstances, the intersection curve might [u]degenerate into curves of lesser degree[/u]. Such a special circumstance is if the interpenetration of the two quadratic surfaces has [u]two double points[/u] (id est the t[u]wo quadratic surfaces are tangent to each other at two points[/u]) or if the two surfaces share a generator (for instance two cones). The separate curves have lesser degree than expected, but their sum will be equal to the product of the degrees of the surfaces. Most famous and important example is when the quartic [u]intersection curve of two quadratic surfaces degenerates into two quadratic curves[/u] (for instance the interpenetration of two circular cylinder might result in two intersecting ellipses: 2+2=4 -> see groin vaults). It might also happen that the result of intersection is a line (linear "curve" of first degree) and a cubic curve (of third degree), like in case of two cones sharing a generator but the apex (1+3=4 -> see [url=https://www.geogebra.org/m/hac8dv7a][u]interactive figure[/u][/url] by Ms. Johanna Pék). Note: the special cases mentioned above are also examples on degenerated intersection curves, just some parts of the curves are mathematically imaginary, thus they are "missing".[/*][/list]
[i]Interpenetration of cylinders whose axes are intersecting each other and the radii of their base circles are equal in length. The [b]i[/b][b]ntersection curve degenerates into two ellipses[/b] (2+2=4). A well known application of this case is the so called groin vault (take only two half-cylinders). [url=https://www.geogebra.org/m/bpm8znk2]Interactive figure[/url][/i]
We will mostly construct interpenetrations which have at least one plane of symmetry. The plane of symmetry is the common plane of symmetry of the two surfaces. We will use [b]image planes parallel to the plane of symmetry[/b] to simplify our construction. If the surfaces do not have a common plane of symmetry, the intersection curve will be asymmetric.[br][br]Note: [u]the image of a symmetric space curve is not necessarily symmetric[/u] (see the figures above which all have a plane of symmetry).[br][br]The plane of symmetry of a...[br][list][*][b]sphere[/b] is each plane passing through its center.[/*][*][b]right circular cylinder[/b] is each plane containing the axis or being perpendicular to it (do not forget: it is an infinite surface).[/*][*][b]right circular cone[/b] is each plane containing the axis or being perpendicular to it and at the same passing through the apex.[/*][/list]
[i]The interpenetration of a right circular cone and a right circular cylinder has a [b]plane of symmetry[/b] if their axes are perpendicular to each other (if the axes of the cylinder passed through apex of the cone, there would be even more planes of symmetry). [url=https://www.geogebra.org/m/ambufgq4]Interactive figure[/url][/i]
During this course, we will regard the plane intersection of a surface simple if...[br][list][*]the intersection consists of either lines or circles.[/*][*]in case of circles, their images remain circles or we can [br]generate such an image easily (for instance by means of a fourth image [br]in Multiview).[/*][/list][br]Sphere[list][*]All its plane intersections are [b]circles[/b].[/*][*]Horizontal (first principal) planes or second principal planes [br](or fourth principal planes) will intersect the surface so, that one of [br]the images will be a circle.[/*][/list][br]Right circular cone[br][list][*]Planes passing through the apex will intersect the surface so that the result is [b]two intersecting lines[/b] (or tangent plane -> one line, plane having a certain angle with the axis -> one point, the apex itself).[/*][*]Planes being perpendicular to the axis will intersect the surface so that the result is a [b]circle[/b] (or if the plane passes through the apex, as well -> one point, the apex itself).[/*][/list][br]Right circular cylinder[br][list][*]Planes being parallel to the axis (or the generators) will intersect the surface so that the result is [b]two parallel[/b][b] lines[/b] (or tangent plane -> one line, other planes -> no intersection).[/*][*]Planes being perpendicular to the axis will intersect the surface so that the result is a [b]circle[/b].[/*][/list]Torus[br][list][*]Planes being perpendicular to the axis will intersect the surface so that the result is [b]two circles[/b] (so called [b]parallels[/b], in case of the two parabolic circles -> only one circle).[/*][*]Planes containing the axis will intersect the surface so that the result is [b]two circles[/b], as well (the so called [b]meridians[/b]).[/*][/list][br]Our constructions will be based on finding such [b]auxiliary planes which intersect both surfaces so that the results are simple intersections[/b].Application of "simple intersections".
[i]Choosing the proper [b]auxiliary planes [/b]in case of a cone and a cylinder. The planes 1) pass through the apex of the cone and 2) are parallel to the generators of the cylinder. Note: in this case we could use horizontal auxiliary planes, as well, since those plane intersections will result in circles (cone) and lines (cylinder). [url=https://www.geogebra.org/m/au8ms43u]Interactive figure[/url][/i]
Sometimes simple intersections can be achieved by intersecting curved surfaces (for instance spheres). The application of [b]auxiliary spheres[/b] makes it possible to construct the [b]interpenetration of surfaces of revolution with intersecting axes[/b] very swift.
[i]Intersection of two surfaces of revolution with intersecting axes. The intersection of the axes is the center of the [b]auxiliary spheres[/b]. If the image plane is parallel to the plane of symmetry, the intersection circles have an edge making it possible to construct the curve very easily. [url=https://www.geogebra.org/m/d3qh3gqy]Interactive figure[/url][br][br]Another example of a cone and cylinder: see [url=https://www.geogebra.org/m/d3jpnvwm][u]interactive figure[/u][/url] by Ms. Johanna Pék Ph.D.[/i]
We will only construct tangents at regular points. [b]At a regular point of the intersection the tangent line is the intersection line of the two tangent planes of the surfaces[/b]. Since a line is determined by two points, and one of the point of this tangent line is the actual intersection point, we only have to find one additional point on the tangent line. Usually, construction of the tracing point of the tangent line as the intersection point of the tracing lines of the tangent planes is the most expedient. [url=https://www.geogebra.org/m/wsj5pnev][u]Interactive 3D figure[/u][/url] (the tangent line as the intersection line of tangent planes).
[i]Detail of the [url=https://www.geogebra.org/m/wsj5pnev]interactive figure[/url]: the red line is the tracing line of the tangent plane of the cylinder, the yellow line is the tracing line of the tangent plane of the cone, their intersection point is the [b]tracing point of the tangent line[/b] at the point we are examining.[/i]
Special cases of tangents: [list][*]In case of [b]singular points[/b], we won't construct the tangents (the upper method does not work anyway, since the two tangent planes are the same in case of the double point, for instance).[/*][*]At [b]regular symmetry points[/b] (id est not the double points), the tangent line is always [b]perpendicular to the plane of symmetry[/b] (in our cases, usually at the same time horizontal, as well).[/*][*]At the [b]topmost and bottommost points[/b] the curve has to "turn back", so the tangent line is always [b]horizontal[/b]. That is true for such intersection points, as well, which are not the topmost or bottommost points regarding the overall intersection, but they are the highest or lowest points in their environment. Try it on the interactive figure: the symmetry points are local topmost and bottommost points, so the tangents will be horizontal.[/*][/list]
We always take care of constructing the following points (similarly to the plane intersections):[br][list][*]points in the [b]symmetry[/b] plane,[/*][*][b]double[/b] points,[/*][*][b]contour[/b] points for both surfaces (in case of Multivew for all the images where contour exists),[/*][*]topmost and bottommost points (many times they are in the symmetry plane),[/*][*]leftmost and rightmost points (if possible),[/*][*]general points (where there is a big gap between two adjecent points).[/*][/list][br]How to connect the points?[br][list][*]In case of Multiview and the cases we examine, the plane of symmetry has an edge view in the top view. That makes the connection easier because the[b] first image[/b] of the space curve will have an actual [b]axis of symmetry[/b].[/*][*]The image of a space curve is a plane curve. The [b]degree of the actual curve and its image is usually the same[/b], id est the image of a quartic space curve will be a quartic plane curve (you can easily check its degree by imagining a line intersecting the curve: a line should intersect a quartic curve at maximum 4 points). The images parallel to the plane of symmetry are exceptions: in that case the degree of the image is half of that of the space curve (think of the coinciding points: every point of this curve is the image of two points coinciding).[/*][*]In case of [b]double points[/b] you can expect a kind of '8' shape, at these double points the intersection curve intersects itself.[/*][/list]
[i][i]If the [b]plane of symmetry[/b] has an [b]edge view[/b], the image of the intersection space curve will have an actual axis of symmetry (for instance: first image in Monge in case of a horizontal cylinder and a vertical cone). [url=https://www.geogebra.org/m/ambufgq4]Interactive figure[/url][/i][/i]
[i][i][i]The [b]special image of a quartic curve[/b] (of 4th degree) can be an arc of a circle which is a quadratic curve (of 2nd degree). In this case the cylinder can be regarded as a projecting cylinder which position makes it very swift to construct intersection points.[/i][/i][/i]
Take care of accuracy:[br][list][*]All the [b]intersection points should be on or inbetween the contour lines or curves of both surfaces[/b]. Only the contour points of the intersection curve may be on the contour, any other points must be inside the boundaries determined by the contours.[/*][*]If you notice an intersection point not abiding the rule above, it shows an unequivocal inaccuracy or mistake in your construction.[/*][/list]
[i][i][i][i]An example: a simplified side view of the interpenetration of a cone and cylinder without the visibility or the intersection curve. All the i[b]ntersection points should lie inside the red trapezium[/b]. The [b]contour points[/b] of the cylinder should lie on one the bases of the trapezium (horizontal red segments), those of the cone should lie on of the legs (left and right sides).[/i][/i][/i][/i]
Visibility[br][list][*]Constructing the contour points is inevitable if we want to represent the interpenetration of two surfaces.[/*][*]The surface whose contour plane (or in case of not quadratic sarfaces contour curve) is in front of the other will cover the other surface.[/*][/list][br]Solid or hollow?[br][list][*]We might represent the known surfaces as 2D curved [b]surfaces which are empty inside[/b]. Usually we only represent the lateral surface of a cone or cylinder. In this case the result of the interpenetration will be a [b]curve on the remaining surface and a hole inside the curve[/b]. You have to take care that we can see through this hole or holes. See: Boolean operations.[/*][*]We might choose to represent them as [b]solids[/b], when we consider not just the surface but everything inside of it, as well. In this case the [b]"negative" imprint of the removed solid will remain inside the remaining solid[/b], so some part of the contour of the removed surface should be also considered.[/*][/list]
[i][i][i][i]An example of a complete interpenetration of a cone and a cylinder, after the removal of the cylinder if the cone is a 1) solid or only a 2) surface[/i][/i][/i][/i]
Boolean operations interpreted in case of interpenetrations[br][list][*][b]Union[/b]: both surfaces remain intact.[/*][*][b]Intersection[/b]: only set of points which belong to both solids.[/*][*][b]Difference[/b]: we only represent one of the surfaces after the removal of the other one.[/*][/list]
[i][i][i][i][i]Boolean operations by H. Potman, A. Asperl, M. Hofer, A. Kilian in Architectural Geometra, Bentley Insitute Press, 2007, p. 113[/i][/i][/i][/i][/i]
[i]An example of the different Boolean operations in case of [b]solid sphere and cube [/b][/i][i] by H. Potman, A. Asperl, M. Hofer, A. Kilian in Architectural Geometra, Bentley Insitute Press, 2007, p. 114[/i]
[i]Some other examples in case of [b]solid cylinder and cone [/b]by József Maklár in Ábrázoló Geometry, Képzőművészeti Kiadó, 1980. [br][br]Note: the auxiliary plane in this case is parallel to the generators of he cylinder and passes through the apex, the same way like we saw before in case of a horizontal cylinder.[/i]
Intersection of surfaces in Multiview[br][br]Let is examine how we can construct the interpenetration of two surfaces in Monge' projection:[br][list][*]Many times the auxiliary planes will be horizontal (first principal planes), the second image has an edge view. In this case we can construct the intersection points directly on the second image and transmit them to the first image. [/*][*]It is useful (in case of tangent surfaces or existing points of symmetry it is actually necessary) to have a fourth image parallel to the plane of symmetry. If the auxiliary planes are not horizontal, they might have an edge view on the fourth image.[/*][*]The first image of the plane of symmetry is usually an edge view which makes it possible to connect the points there easier, since the image of the intersection curve will have an actual axis of symmetry.[/*][/list]
[i][i]An example of interpenetration (special case: the second image plane is parallel to the plane of symmetry). [url=https://www.geogebra.org/m/gvpz2zd3]I[/url][u][url=https://www.geogebra.org/m/gvpz2zd3]nteractive figure[/url][/u] by Ms. Johanna Pék Ph.D.[/i][/i]
[i][i][i]Another example if the position of the cylinder is not special. [url=https://www.geogebra.org/m/dz9xmtpn]I[/url][u][url=https://www.geogebra.org/m/dz9xmtpn]nteractive figur[/url][/u][u][url=https://www.geogebra.org/m/dz9xmtpn]e[/url][/u] by Ms. Johanna Pék Ph.D.[/i][/i][/i]
[i]Intersection of a sphere and a right circular cone.[/i]
Intersection of surfaces in frontal axonometry[br][br]During the practical class, we will construct the interpenetration of a cone and cylinder in frontal axonometry. Remember the second image (id est the orthogonal projection on [x,z] coordinate plane) and the rotated [x,y] coordinate plane can be altogether regarded as the second and first image of a Monge' projection.
[i]An example in frontal axonometry. [url=https://www.geogebra.org/m/r4ncmyhw]Interactive figure[/url][/i]
[br][img width=250,height=250]https://edu.epitesz.bme.hu/pluginfile.php/36277/mod_page/content/40/csovek_1.jpg[/img] [img width=205,height=250]https://edu.epitesz.bme.hu/pluginfile.php/36277/mod_page/content/40/csovek_2.jpg[/img] [img width=250,height=250]https://edu.epitesz.bme.hu/pluginfile.php/36277/mod_page/content/40/csovek_3.jpg[/img][br][i]Pipe connections. What kind of surfaces and intersection curves can you recognize?[/i][br][br][br][i][img width=300,height=375]https://edu.epitesz.bme.hu/pluginfile.php/36277/mod_page/content/40/13.jpeg[/img][br]Bubbles as intersection of spheres[/i][br][br][br][i][img width=267,height=401]https://edu.epitesz.bme.hu/pluginfile.php/36277/mod_page/content/40/Velence%20Santa%20Maria%20dei%20Carmini.jpg[/img] [img width=390,height=400]https://edu.epitesz.bme.hu/pluginfile.php/36277/mod_page/content/40/V%C3%A1c%20sz%C3%A9kesegyh%C3%A1z%201.JPG[/img][br]Vaults as types of intersections of cylinders: Venice, Santa Maria dei Carmini | Vác, Cathedral[/i][br][br][br][i][img width=300,height=300]https://edu.epitesz.bme.hu/pluginfile.php/36277/mod_page/content/40/R%C3%B3ma%20Pantheon%202.jpg[/img][img width=429,height=300]https://edu.epitesz.bme.hu/pluginfile.php/36277/mod_page/content/40/R%C3%B3ma%20Pantheon%203.jpg[/img][br]Parthenon, Rome. The light rays form a cylinder, which intersects the dome (sphere) along another circle. This is a special case of degenerated intersection curves without double points. Note: circles are quadratic, so altogether the degree of the intersection curve is 2+2=4.[/i][br][br][br][i][img width=333,height=250]https://edu.epitesz.bme.hu/pluginfile.php/36277/mod_page/content/40/Lisszabon%20metr%C3%B3%20Baixa-Chiado%201.jpg[/img] [img width=333,height=250]https://edu.epitesz.bme.hu/pluginfile.php/36277/mod_page/content/40/Lisszabon%20metr%C3%B3%20Baixa-Chiado%202.jpg[/img][br][/i][i]Lisbon, underground station Baixa-Chiado by Alvaro Siza[/i][br]
