DOCSLIB.ORG

Interaction with 3D Geometry

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Interaction with 3D Geometry Interaction With 3D Geometry Stan Melax Graphics Software Engineer, Intel 3D Interaction Happens with Geometric Objects Rigid Body Skinned Characters Soft Body { { { Mesh geometry; Mesh geometry; Mesh geometry; Vec3 position; Vec3 position[]; Springs connectivity; Quat orientation; Quat orientation[]; }; }; }; Agenda – Interacting with 3D Geometry Practical topics among: ● Core Geometry Concepts ● Convex Polyhedra ● Spatial and Mass Properties ● Soft Geometry and Springs With examples and implementation issues. Warning: Some Math Ahead 푚 푚 푚 푣 v 푢 × 푣 00 01 02 푥 푀푣 = 푚10 푚11 푚12 푣푦 v 푚 푚 푚 푣 u 20 21 22 푧 u v 푢푥푣푥 푢푥푣푦 푢푥푣푧 푢 ∙ 푣 > 0 푢푣푇 = 푢푦푣푥 푢푦푣푦 푢푦푣푧 u u 푢푧푣푥 푢푧푣푦 푢푧푣푧 v 휕푓푥 휕푓푥 휕푓푥 v 휕푣푥 휕푣푦 휕푣푧 u v 푑푓 휕푓푦 휕푓푦 휕푓푦 = 푑푣 휕푣 휕푣 휕푣 u 푥 푦 푧 b 휕푓푧 휕푓푧 휕푓푧 푢 ∙ 푣 < 0 a 푣 × 푢 휕푣푥 휕푣푦 휕푣푧 Vector Arithmetic Dot Product Cross Product Matrix Stuff What’s an “outer product”? Familiar dot or inner product: 푣푥 푇 푢 푢 푢 푢 ∙ 푣 = 푢 푣 = 푥 푦 푧 푣푦 = 푢푥푣푥 + 푢푦푣푦 + 푢푧푣푧 푣푧 Outer product: 푢푥 푢푥푣푥 푢푥푣푦 푢푥푣푧 푇 푢⨂푣 = 푢푣 = 푢푦 푣푥 푣푦 푣푧 = 푢푦푣푥 푢푦푣푦 푢푦푣푧 푢푧 푢푧푣푥 푢푧푣푦 푢푧푣푧 Outer Product - Geometric View v v v u u u 푢 = 1 푢 ∙ 푣 푢(푢 ∙ 푣) Distance of v along u (scalar) Projection of v along u (vector) Outer product 풖 ⊗ 풖 of a 푢 푢 ∙ 푣 ≠ 푢 ∙ 푢 푣 unit vector 풖 projects any vector along that direction. 푢 푢 ∙ 푣 = 푢 ⊗ 푢 푣 Outer Product for Plane Projection −푢 푢 ∙ 푣 −푢 푢 ∙ 푣 v v 푣 − 푢 푢 ∙ 푣 v u u 푢 푢 ∙ 푣 u Remove portion of v or (푢 ⊗ 푢)푣 that runs along u 푣 − 푢 푢 ∙ 푣 = 퐼푣 − 푢 ⊗ 푢 푣 = (퐼 − 푢 ⊗ 푢 )푣 푰 − 풖 ⊗ 풖 projects any vector onto plane with normal 풖. Numerical Precision Issues 3 “collinear” points Triangles and Planes Could use: ● Specify Front and Back 퐴푥 + 퐵푦 + 퐶푧 == 퐷 Prefer convention: 퐴푥 + 퐵푦 + 퐶푧 + 퐷 == 0 c 0 0 0 푝푤 Given a point [xyz] its 푝푥 푝푦 푝푧 distance above plane is: a b 푝푥 푥 + 푝푦 푦 + 푝푧 푧 + 푝푤 • <0 below Triangle (푎 푏 푐) Plane 푝 = 푝 푝 푝 푝 푥 푦 푧 푤 • =0 on plane CCW winding • >0 above Intersection of 3 planes p0 p2 푝0푥 푣푥 + 푝0푦 푣푦 + 푝0푧 푣푧 + 푝0푤 == 0 p1 푝1푥 푣푥 + 푝1푦 푣푦 + 푝1푧 푣푧 + 푝1푤 == 0 푝2푥 푣푥 + 푝2푦 푣푦 + 푝2푧 푣푧 + 푝2푤 == 0 p0 푝0푥 푝0푦 푝0푧 푝0푤 푣푥 푃 = 푝1푥 푝1푦 푝1푧 , 푏 = 푝1푤 , 푣 = 푣푦 푝2 푝2 푝2 푝2 푣푧 푥 푦 푧 푤 v 푃푣 = −푏 p1 p2 푣 = −푃−1푏 What if we have 4 planes? Example: Floor Only Walls Roof Planes Roof Planes overhead view As house grows bottom up, how will the 4 roof planes come together at the top of this house? Determining How 4 Planes Meet Note: these are top down views p3 of a convex region p0 p2 ? p1 푝0 푝0 If 4 planes meet at a point xyz 푥 푦 푝0푧 푝0푤 푥 0 then we’ve found a solution to: 푝1푥 푝1푦 푝1푧 푝1푤 푦 0 푧 = 푝2푥 푝2푦 푝2푧 푝2푤 0 1 푝3푥 푝3푦 푝3푧 푝3푤 0 Only possible if matrix singular. Determining How 4 Planes Meet p3 p0 p2 ? p1 d<0 d==0 d>0 푝0푥 푝1푥 푝2푥 푝3푥 푝0푦 푝1푦 푝2푦 푝3푦 Notes: these are top down views of a convex region. 푝0 푝1 푝2 푝3 d = = Planes well “behaved” all point same way. 푝0푧 푝1푧 푝2푧 푝3푧 Planes in CCW order. 푝0푤 푝1푤 푝2푤 푝3푤 Det of any 3x3 subblock from first 3 xyz rows is >0. Intersect Line Plane 푝푥푦푧 ∙ 푣푡 + 푝푤 푝푥푦푧 v1 푑1 = 푝푥푦푧 ∙ 푣1 +푝푤 풅ퟏ p time 0 1 푝 푝 푝 t=? 푝푥푦푧 = 푥 푦 푧 v0 푑0 = 푝푥푦푧 ∙ 푣0 +푝푤 풅ퟎ 푑푖푠푡 = 푝푤 distance above plane distance above −푑 푝푥푦푧 ∙ 푣0 + 푝푤 푖푚푝푎푐푡 = 푣 + 푣 − 푣 푡 푡 = 0 = − 0 1 0 (푝푥푦푧 ∙ 푣1 + 푝푤) − (푝푥푦푧 ∙ 푣0 + 푝푤) 푑1 − 푑0 Simple Ray Triangle Intersection Test ● Intersect ray with plane and get a point p. p ● For each edge va to vb ● Cross product with p-va ● Dot result with tri normal n ● <0 means outside ● Backside hit if dot(n,ray)>0 Ray Mesh Intersection ● Check Against Every Polygon ● (spatial structures can rule out many quickly) Multiple impact points – take closest. ● Detecting a “hit” - might not be closest ● Numerical Robustness – don’t slip between two adjacent triangles. ● Mesh “intact” – t-intersections, holes Numeric precision can result in plane intersection point landing just caused by missing triangles. outside each time, thus missing both neighbors. Solid Geometry ● “Water-Tight” borderless manifold mesh: ● Every edge has 1 adjacency edge going other way ● Consistent winding. Polygon normals all face to exterior. ● The mesh is the boundary representation - the infinitely thin surface that separates solid matter from empty space. Inside/Outside Test of a point ● Cast a long ray from point ray ● point is inside if it first hits the backside of a polygon. ● point is outside the object if it first hits a front side or p nothing at all. Convex Polyhedra Convex Mesh 풂 풃 Math textbook: 푲 퐾 푐표푛푣푒푥 ↔ ∀푎, 푏 ∈ 퐾 → 푎푏 ⊆ 퐾 ● Neighboring face normals tilt away. ● Volume bounded by a number of planes. ● Every vertex lies at or below every face. Many algorithms much easier when using convex shapes instead of general meshes. Convex In/Out test A point is inside a convex volume if it lies under every plane boundary. p w No testing with vertices or edges. q Convex Ray Intersection v0 v0 v0 v1 v0 ● Crop Ray with all front facing planes: 푛 ∙ 푣1 − 푣0 < 0 ● Impact at v0 if under all backside planes Convex Line-Segment Intersection v0 v0 v1 v1 v0 v1 ● Trim v0 for Front facing planes 푛 ∙ 푣1 − 푣0 < 0 ● Trim v1 for Back facing planes 푛 ∙ 푣1 − 푣0 > 0 Convex-Convex Contact ● Separating Axis Theorem – Two non touching convex polyhedra are separated by a plane. ● The contact between two touching convex polyhedra will be either ● A point ● A line segment ● A convex polygon Physics engines like convex objects Convex Hull from points Convex Hull - smallest convex polyhedron that contains all the points. ● Convex hull of a mesh will contain the mesh. ● Often a sufficient proxy for interaction and collision Techniques ● Expanding outward: QuickHull [Eddy ’77] ● Reducing inward: Gift Wrapping [Chand ’70] Compute 3D Convex Hull ● Start with 2 back to back triangles. (or a tetrahedron) ● Find a vertex outside current volume (above faces). ● Find edge loop silhouette (around all faces below point) ● Replace with new fan of polys ● Remove Folds (if any) ● Rinse and Repeat Hull Numerical Robustness Grow Hull Generated skinny Flip edge to fix triangle with bad normal. Hull Tri-Tet Implementation ● Simple Triangle-mesh based approach ● When point d above any [abc] add tetrahedron [abcd] (triangles [acb][dab][dbc][dca]) to mesh. ● Prune any back-to-back tris such as [abc] and [bac] d d d a c a c a c b b b Hull Tri-Tet Numeric Robustness a a a a c c e c e c e e b d d b d b d b ● 5 points {a,b,c,d,e} are coplanar-ish at one end of the point cloud ● But next point e tests above triangle [abc] but below [adb]. ● Silhouette is {abc} for which we extrude a new tetrahedron up to e ● This produces triangles [eca],[ebc] (blue) facing the right way and [eab] (red) facing the wrong way – based on known interior point. ● This provides the hindsight to see that [adb] should also be extruded Simplified Convex Hull ● Off-the-shelf solutions typically generate the complete hull. ● All hull vertices may not be needed and may be inefficient for usage. ● Instead use greedy incremental algorithm picking next vertex that increases hull size the most. ● Stop when vertex count or error Full Hull Simplified threshold reached Minimize Number of Planes vs Points Minimize Planes: ● Compute full hull 푝푥 푝푦 푝푧 ● Dual points 푝푤 푝푤 푝푤 ● Compute simplified hull 12 Vertices 20 Vertices ● Take Dual 20 Planes 12 Planes Convex Decomposition 3DS Max Screenshot Manual Creation of Automatic Mesh Use skinned mesh Collision Proxy Decomposition bone weights to [Ratcliff], [Mamou] create collision hulls Example: Convex Bones For Collisions Ragdolls on Stairs Hand catching coins Constructive Solid Geometry Boolean Operations Solid Geometry Boolean Operations Applications for Booleans Art tools (already have CSG) ● Modeling and level design Game Usages (less fidelity required): ● Destruction ● Geomod (tunneling) A convex based approach may suffice. Convex Cropping and Intersection ● Like general mesh cropping, but only 1 open loop that is itself a convex polygon. ● Intersecting two convexes can be done by cropping one with all the planes of the other ● Non-convex operands can be implemented as a union Convex Cropping Robustness ● Floating point issues can occur even in this most basic of mesh operations. ● One solution is to utilize plane equations to determine face/plane connectivity. Bad Slice Correct Slice Pre Slice Robustness with Quantum Planes Let all planes p of CSG Boolean operands satisfy: 푠 푝 푠 푝 ∀푝∃푠 ∶ 푠푝 = 푥 푦 푠 푝푧 푠 푝푤 , 푠 푝{푥,푦,푧,푤} ∈ ℤ, 푠푝{푥,푦,푧} ≤ 푘 푠 푝0 푠 푝0 푠 푝0 0 푥 0 푦 0 푧 푠0푝0푤 Recall how vertices are the −1 푀 = 푠1푝1푥 푠1푝1푦 푠1푝1푧 푣 = −푀 푠1푝1푤 intersection of 3 or more planes 푠2푝2푥 푠2푝2푦 푠2푝2푧 푠2푝2푤 푎 ∃푎, 푏: 푣 = , 푎, 푏 ∈ ℤ , 푏 ≤ det 푀 ≤ 6푘3 Generated points have {푥,푦,푧} 푏 Finite denominator ∴ ∃휀∀푢, 푣: 푢 − 푣 < 휀 ↔ 푢 = 푣 A fixed epsilon can now be used to indicate if two points are the same.
Load more

Recommended publications
  • The Apollonius Problem on the Excircles and Related Circles
    Forum Geometricorum b Volume 6 (2006) xx–xx. bbb FORUM GEOM ISSN 1534-1178 The Apollonius Problem on the Excircles and Related Circles Nikolaos Dergiades, Juan Carlos Salazar, and Paul Yiu Abstract. We investigate two interesting special cases of the classical Apollo- nius problem, and then apply these to the tritangent of a triangle to find pair of perspective (or homothetic) triangles. Some new triangle centers are constructed. 1. Introduction This paper is a study of the classical Apollonius problem on the excircles and related circles of a triangle. Given a triad of circles, the Apollonius problem asks for the construction of the circles tangent to each circle in the triad. Allowing both internal and external tangency, there are in general eight solutions. After a review of the case of the triad of excircles, we replace one of the excircles by another (possibly degenerate) circle associated to the triangle. In each case we enumerate all possibilities, give simple constructions and calculate the radii of the circles. In this process, a number of new triangle centers with simple coordinates are constructed. We adopt standard notations for a triangle, and work with homogeneous barycen- tric coordinates. 2. The Apollonius problem on the excircles 2.1. Let Γ=((Ia), (Ib), (Ic)) be the triad of excircles of triangle ABC. The sidelines of the triangle provide 3 of the 8 solutions of the Apollonius problem, each of them being tangent to all three excircles. The points of tangency of the excircles with the sidelines are as follows. See Figure 2. BC CA AB (Ia) Aa =(0:s − b : s − c) Ba =(−(s − b):0:s) Ca =(−(s − c):s :0) (Ib) Ab =(0:−(s − a):s) Bb =(s − a :0:s − c) Cb =(s : −(s − c):0) (Ic) Ac =(0:s : −(s − a)) Bc =(s :0:−(s − c)) Cc =(s − a : s − b :0) Publication Date: Month day, 2006.
    [Show full text]
  • Stanley Rabinowitz, Arrangement of Central Points on the Faces of A
    International Journal of Computer Discovered Mathematics (IJCDM) ISSN 2367-7775 ©IJCDM Volume 5, 2020, pp. 13{41 Received 6 August 2020. Published on-line 30 September 2020 web: http://www.journal-1.eu/ ©The Author(s) This article is published with open access1. Arrangement of Central Points on the Faces of a Tetrahedron Stanley Rabinowitz 545 Elm St Unit 1, Milford, New Hampshire 03055, USA e-mail: [email protected] web: http://www.StanleyRabinowitz.com/ Abstract. We systematically investigate properties of various triangle centers (such as orthocenter or incenter) located on the four faces of a tetrahedron. For each of six types of tetrahedra, we examine over 100 centers located on the four faces of the tetrahedron. Using a computer, we determine when any of 16 con- ditions occur (such as the four centers being coplanar). A typical result is: The lines from each vertex of a circumscriptible tetrahedron to the Gergonne points of the opposite face are concurrent. Keywords. triangle centers, tetrahedra, computer-discovered mathematics, Eu- clidean geometry. Mathematics Subject Classification (2020). 51M04, 51-08. 1. Introduction Over the centuries, many notable points have been found that are associated with an arbitrary triangle. Familiar examples include: the centroid, the circumcenter, the incenter, and the orthocenter. Of particular interest are those points that Clark Kimberling classifies as \triangle centers". He notes over 100 such points in his seminal paper [10]. Given an arbitrary tetrahedron and a choice of triangle center (for example, the circumcenter), we may locate this triangle center in each face of the tetrahedron. We wind up with four points, one on each face.
    [Show full text]
  • Searching for the Center
    Searching For The Center Brief Overview: This is a three-lesson unit that discovers and applies points of concurrency of a triangle. The lessons are labs used to introduce the topics of incenter, circumcenter, centroid, circumscribed circles, and inscribed circles. The lesson is intended to provide practice and verification that the incenter must be constructed in order to find a point equidistant from the sides of any triangle, a circumcenter must be constructed in order to find a point equidistant from the vertices of a triangle, and a centroid must be constructed in order to distribute mass evenly. The labs provide a way to link this knowledge so that the students will be able to recall this information a month from now, 3 months from now, and so on. An application is included in each of the three labs in order to demonstrate why, in a real life situation, a person would want to create an incenter, a circumcenter and a centroid. NCTM Content Standard/National Science Education Standard: • Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships. • Use visualization, spatial reasoning, and geometric modeling to solve problems. Grade/Level: These lessons were created as a linking/remembering device, especially for a co-taught classroom, but can be adapted or used for a regular ed, or even honors level in 9th through 12th Grade. With more modification, these lessons might be appropriate for middle school use as well. Duration/Length: Lesson #1 45 minutes Lesson #2 30 minutes Lesson #3 30 minutes Student Outcomes: Students will: • Define and differentiate between perpendicular bisector, angle bisector, segment, triangle, circle, radius, point, inscribed circle, circumscribed circle, incenter, circumcenter, and centroid.
    [Show full text]
  • Arxiv:2101.02592V1 [Math.HO] 6 Jan 2021 in His Seminal Paper [10]
    International Journal of Computer Discovered Mathematics (IJCDM) ISSN 2367-7775 ©IJCDM Volume 5, 2020, pp. 13{41 Received 6 August 2020. Published on-line 30 September 2020 web: http://www.journal-1.eu/ ©The Author(s) This article is published with open access1. Arrangement of Central Points on the Faces of a Tetrahedron Stanley Rabinowitz 545 Elm St Unit 1, Milford, New Hampshire 03055, USA e-mail: [email protected] web: http://www.StanleyRabinowitz.com/ Abstract. We systematically investigate properties of various triangle centers (such as orthocenter or incenter) located on the four faces of a tetrahedron. For each of six types of tetrahedra, we examine over 100 centers located on the four faces of the tetrahedron. Using a computer, we determine when any of 16 con- ditions occur (such as the four centers being coplanar). A typical result is: The lines from each vertex of a circumscriptible tetrahedron to the Gergonne points of the opposite face are concurrent. Keywords. triangle centers, tetrahedra, computer-discovered mathematics, Eu- clidean geometry. Mathematics Subject Classification (2020). 51M04, 51-08. 1. Introduction Over the centuries, many notable points have been found that are associated with an arbitrary triangle. Familiar examples include: the centroid, the circumcenter, the incenter, and the orthocenter. Of particular interest are those points that Clark Kimberling classifies as \triangle centers". He notes over 100 such points arXiv:2101.02592v1 [math.HO] 6 Jan 2021 in his seminal paper [10]. Given an arbitrary tetrahedron and a choice of triangle center (for example, the circumcenter), we may locate this triangle center in each face of the tetrahedron.
    [Show full text]
  • Triangle-Centers.Pdf
    Triangle Centers Maria Nogin (based on joint work with Larry Cusick) Undergraduate Mathematics Seminar California State University, Fresno September 1, 2017 Outline • Triangle Centers I Well-known centers F Center of mass F Incenter F Circumcenter F Orthocenter I Not so well-known centers (and Morley's theorem) I New centers • Better coordinate systems I Trilinear coordinates I Barycentric coordinates I So what qualifies as a triangle center? • Open problems (= possible projects) mass m mass m mass m Centroid (center of mass) C Mb Ma Centroid A Mc B Three medians in every triangle are concurrent. Centroid is the point of intersection of the three medians. mass m mass m mass m Centroid (center of mass) C Mb Ma Centroid A Mc B Three medians in every triangle are concurrent. Centroid is the point of intersection of the three medians. Centroid (center of mass) C mass m Mb Ma Centroid mass m mass m A Mc B Three medians in every triangle are concurrent. Centroid is the point of intersection of the three medians. Incenter C Incenter A B Three angle bisectors in every triangle are concurrent. Incenter is the point of intersection of the three angle bisectors. Circumcenter C Mb Ma Circumcenter A Mc B Three side perpendicular bisectors in every triangle are concurrent. Circumcenter is the point of intersection of the three side perpendicular bisectors. Orthocenter C Ha Hb Orthocenter A Hc B Three altitudes in every triangle are concurrent. Orthocenter is the point of intersection of the three altitudes. Euler Line C Ha Hb Orthocenter Mb Ma Centroid Circumcenter A Hc Mc B Euler line Theorem (Euler, 1765).
    [Show full text]
  • The Feuerbach Point and Euler Lines
    Forum Geometricorum b Volume 6 (2006) 191–197. bbb FORUM GEOM ISSN 1534-1178 The Feuerbach Point and Euler lines Bogdan Suceav˘a and Paul Yiu Abstract. Given a triangle, we construct three triangles associated its incircle whose Euler lines intersect on the Feuerbach point, the point of tangency of the incircle and the nine-point circle. By studying a generalization, we show that the Feuerbach point in the Euler reflection point of the intouch triangle, namely, the intersection of the reflections of the line joining the circumcenter and incenter in the sidelines of the intouch triangle. 1. A MONTHLY problem Consider a triangle ABC with incenter I, the incircle touching the sides BC, CA, AB at D, E, F respectively. Let Y (respectively Z) be the intersection of DF (respectively DE) and the line through A parallel to BC.IfE and F are the midpoints of DZ and DY , then the six points A, E, F , I, E, F are on the same circle. This is Problem 10710 of the American Mathematical Monthly with slightly different notations. See [3]. Y A Z Ha F E E F I B D C Figure 1. The triangle Ta and its orthocenter Here is an alternative solution. The circle in question is indeed the nine-point circle of triangle DY Z. In Figure 1, ∠AZE = ∠CDE = ∠CED = ∠AEZ. Therefore AZ = AE. Similarly, AY = AF . It follows that AY = AF = AE = AZ, and A is the midpoint of YZ. The circle through A, E, F , the midpoints of the sides of triangle DY Z, is the nine-point circle of the triangle.
    [Show full text]
  • Degree of Triangle Centers and a Generalization of the Euler Line
    Beitr¨agezur Algebra und Geometrie Contributions to Algebra and Geometry Volume 51 (2010), No. 1, 63-89. Degree of Triangle Centers and a Generalization of the Euler Line Yoshio Agaoka Department of Mathematics, Graduate School of Science Hiroshima University, Higashi-Hiroshima 739–8521, Japan e-mail: [email protected] Abstract. We introduce a concept “degree of triangle centers”, and give a formula expressing the degree of triangle centers on generalized Euler lines. This generalizes the well known 2 : 1 point configuration on the Euler line. We also introduce a natural family of triangle centers based on the Ceva conjugate and the isotomic conjugate. This family contains many famous triangle centers, and we conjecture that the de- gree of triangle centers in this family always takes the form (−2)k for some k ∈ Z. MSC 2000: 51M05 (primary), 51A20 (secondary) Keywords: triangle center, degree of triangle center, Euler line, Nagel line, Ceva conjugate, isotomic conjugate Introduction In this paper we present a new method to study triangle centers in a systematic way. Concerning triangle centers, there already exist tremendous amount of stud- ies and data, among others Kimberling’s excellent book and homepage [32], [36], and also various related problems from elementary geometry are discussed in the surveys and books [4], [7], [9], [12], [23], [26], [41], [50], [51], [52]. In this paper we introduce a concept “degree of triangle centers”, and by using it, we clarify the mutual relation of centers on generalized Euler lines (Proposition 1, Theorem 2). Here the term “generalized Euler line” means a line connecting the centroid G and a given triangle center P , and on this line an infinite number of centers lie in a fixed order, which are successively constructed from the initial center P 0138-4821/93 $ 2.50 c 2010 Heldermann Verlag 64 Y.
    [Show full text]
  • The Feuerbach Point and the Fuhrmann Triangle
    Forum Geometricorum Volume 16 (2016) 299–311. FORUM GEOM ISSN 1534-1178 The Feuerbach Point and the Fuhrmann Triangle Nguyen Thanh Dung Abstract. We establish a few results on circles through the Feuerbach point of a triangle, and their relations to the Fuhrmann triangle. The Fuhrmann triangle is perspective with the circumcevian triangle of the incenter. We prove that the perspectrix is the tangent to the nine-point circle at the Feuerbach point. 1. Feuerbach point and nine-point circles Given a triangle ABC, we consider its intouch triangle X0Y0Z0, medial trian- gle X1Y1Z1, and orthic triangle X2Y2Z2. The famous Feuerbach theorem states that the incircle (X0Y0Z0) and the nine-point circle (N), which is the common cir- cumcircle of X1Y1Z1 and X2Y2Z2, are tangent internally. The point of tangency is the Feuerbach point Fe. In this paper we adopt the following standard notation for triangle centers: G the centroid, O the circumcenter, H the orthocenter, I the incenter, Na the Nagel point. The nine-point center N is the midpoint of OH. A Y2 Fe Y0 Z1 Y1 Z0 H O Z2 I T Oa B X0 U X2 X1 C Ja Figure 1 Proposition 1. Let ABC be a non-isosceles triangle. (a) The triangles FeX0X1, FeY0Y1, FeZ0Z1 are directly similar to triangles AIO, BIO, CIO respectively. (b) Let Oa, Ob, Oc be the reflections of O in IA, IB, IC respectively. The lines IOa, IOb, IOc are perpendicular to FeX1, FeY1, FeZ1 respectively. Publication Date: July 25, 2016. Communicating Editor: Paul Yiu. 300 T. D. Nguyen Proof.
    [Show full text]
  • Key Learning(S): Unit Essential Question(S)
    Student Learning Map Topic: Geometry Gallery Unit 3 Course: Mathematics 1 Key Learning(s): Unit Essential Question(s): 1. The interior and exterior angles of a polygon can be determined by In what real life situations would it be necessary Optional the number of sides of the polygon. to determine the interior or exterior angles of a Instructional Tools: 2. There are special inequalities that exist between the sides of a polygon? Mira, patty paper, triangle, the angles of a triangle, and between angles and their opposite How could the points of concurrency be used to Geometer’s sides. solve a real life situation? Sketchpad, compass, 3. Congruency of triangles can be determined by special relationships What relationships between sides and angles can straight edge, of the sides and angles. be used to prove the congruency of triangles? protractor, 4. Triangles have four different points of concurrency or points of graphing calculator center. How are the parallelogram, rectangle, rhombus, square, trapezoid, and kite alike and different? 5. Special quadrilaterals are related by their properties. Concept: Concept: Concept: Concept: Concept: Measures of interior and Triangular Inequalities Triangle Points of Congruency Theorems Relationships Between exterior angles of Concurrency for Triangles Special Quadrilaterals polygons Lesson Essential Questions Lesson Essential Questions Lesson Essential Questions 1. What does the sum of two sides of Lesson Essential Questions a triangle tell me about the third side? Lesson Essential Questions 1. How do I find points of 1. What characteristics 1. How do I find the sum of 2. In a triangle, what is the the measures of the interior relationship of an angle of a triangle concurrency in triangles? 1.
    [Show full text]
  • Centers of Triangles Circumcenter and Incenter Worksheet
    Centers Of Triangles Circumcenter And Incenter Worksheet Uncomely Casey wreaks some Guatemalan after unsolaced Gavin splining sunwards. Is Tarrant elemental or frayed after ghoulish Hakim indues so versatilely? Reggie cocks toilsomely if graceless Manish sawed or apologising. Are likely sure you still to delete your template? Pythagorean spiral on triangle centers of and triangles circumcenter and length. Which three or incenter, turn text on file from all families should meet. Abis stand for notebooks or incenter of each side of concurrency. Similar Triangles Math Test! Inferno Cantos VI And VII: Quiz! Then be outside of circumcenters, h and intrinsic connections between common core, circumcenter or proofs or subsets of where households in all angle bisectors? Construct a circumcenter is equidistant from center of one bcr and incenter of concurrency of concurrency of a triangle centers of concurrency of a color with respect to continue withsteps and midsegments. Record your students understand this website and select the incenter of triangles and circumcenter is a link to find among the triangle and the license for helping us keep this new triangle? Place point M on the horizontal line. Select name of triangle centers of each of each group member will be in addition to suspend a close up. How would like to access the centers of and triangles from all agree that is the special relationships. Use to your description must be shared with side. Students will explore angle bisectors to snapshot the incenter of current triangle to use perpendicular bisectors to infuse the circumcenter of outer triangle. Point circle that divides an angle bisector, as a iangle.
    [Show full text]
  • SOT: Compact Representation for Triangle and Tetrahedral Meshes
    SOT: Compact Representation for Triangle and Tetrahedral Meshes Topraj Gurung and Jarek Rossignac School of Interactive Computing, College of Computing, Georgia Institute of Technology, Atlanta, GA ABSTRACT The Corner Table (CT) represents a triangle mesh by storing 6 integer references per triangle (3 vertex references in the Vertex table and 3 references to opposite corners in the Opposite table, which accelerate access to adjacent triangles). The Compact Half Face (CHF) representation extends CT to tetrahedral meshes, storing 8 references per tetrahedron (4 in the Vertex table and 4 in the Opposite table). We use the term Vertex Opposite Table (VOT) to refer to both CT and CHF and propose a sorted variation, SVOT, which is inspired by tetrahedral mesh encoding techniques and which works for both triangle and tetrahedral meshes. The SVOT does not require additional storage and yet provides, for each vertex, a reference to an incident corner from which the star (incident cells) of the vertex may be traversed at a constant cost per visited element. We use the corner operators for querying and traversing the triangle meshes while for tetrahedral meshes, we propose a set of powerful wedge-based operators. Improving on the SVOT, we propose our Sorted Opposite Table (SOT) variation, which eliminates the Vertex table completely and hence reduces storage requirements by 50% to only 3 references per triangle for triangle meshes and 4 references and 9 bits per tetrahedron for tetrahedral meshes, while preserving the vertex-to-incident- corner references and supporting the corner operators and our wedge operators with a constant average cost.
    [Show full text]
  • Bicentric Pairs of Points and Related Triangle Centers
    Forum Geometricorum b Volume 3 (2003) 35–47. bbb FORUM GEOM ISSN 1534-1178 Bicentric Pairs of Points and Related Triangle Centers Clark Kimberling Abstract. Bicentric pairs of points in the plane of triangle ABC occur in con- nection with three configurations: (1) cevian traces of a triangle center; (2) points of intersection of a central line and central circumconic; and (3) vertex-products of bicentric triangles. These bicentric pairs are formulated using trilinear coordi- nates. Various binary operations, when applied to bicentric pairs, yield triangle centers. 1. Introduction Much of modern triangle geometry is carried out in in one or the other of two homogeneous coordinate systems: barycentric and trilinear. Definitions of triangle center, central line, and bicentric pair, given in [2] in terms of trilinears, carry over readily to barycentric definitions and representations. In this paper, we choose to work in trilinears, except as otherwise noted. Definitions of triangle center (or simply center) and bicentric pair will now be briefly summarized. A triangle center is a point (as defined in [2] as a function of variables a, b, c that are sidelengths of a triangle) of the form f(a, b, c):f(b, c, a):f(c, a, b), where f is homogeneous in a, b, c, and |f(a, c, b)| = |f(a, b, c)|. (1) If a point satisfies the other defining conditions but (1) fails, then the points Fab := f(a, b, c):f(b, c, a):f(c, a, b), Fac := f(a, c, b):f(b, a, c):f(c, b, a) (2) are a bicentric pair.
    [Show full text]