Computer Graphics Basic 3D Geometry CO2409 Computer Graphics Week 5-1

Today’s Lecture 1.From 2D to 3D Geometry 2.Handedness 3.Meshes 4.Face and Vertex Normals

From 2D to 3D Geometry 3D geometry adds an extra axis over 2D geometry –This “Z” axis represents “depth” Can choose the direction to point Z Giving us the handedness of the 3D axes: –Use thumb (X), index finger (Y) & middle finger (Z) to represent the axes –Use your left hand and the axes are left-handed, otherwise they are right- handed –We will generally use left-handed axes LH RH

Handedness + Definitions DirectX documentation and several graphics applications use left-handed axes OpenGL documentation + maths texts use right-handed Neither is better, just a choice –Note that DirectX and OpenGL can both use either handedness. Handedness is more an issue when drawing diagrams or creating mesh geometry. Defining geometry in 3D is just like 2D Vertices – we add a third coordinate (Z): A(10, 20,15), B(30, 30,20) Edges & Polygons: –No change in the definition from 2D

Meshes A mesh is a set of polygons representing a 3D object Characteristics of meshes: –Polygons connected? –No holes in the polygon surface? –Polygons form a closed solid rather than a surface (i.e. is there an interior and an exterior)? All these types can be rendered –Latter two must be satisfied for an object to appear truly solid.

Triangular Meshes Polygons in a mesh can have any number of sides However, usually all polygons in a mesh are triangles: Graphics hardware tends to support triangle rendering Triangles are always planar –[Planar = flat] –Polygons with four or more edges can be “folded” –Making rendering more complex So polygons are usually split into triangles first

Dividing Polygons into Triangles [Reminder] Convex polygons: –No indentations –Can join any pair of vertices without going outside of polygon Otherwise concave –Split into convex polygons first Polygons can always be split into triangles as shown Multiple simple approaches for convex polygons

Face Normals A mesh is often defined together with a set of face and/or vertex normals A face (polygon) normal is a normalised vector at right angles to a polygon in the mesh Together with the vertices of the polygon this defines the plane of the polygon Face normals generally point outwards –Defining the outside of the mesh surface Face normals can be calculated from polygon edges –Use a math operation called the Cross Product –Will see this in more detail later

Vertex Normals Vertex normals can be defined for each vertex in a mesh They can be calculated: –Calculate the face normals –Average all face normals of the polygons containing the vertex Or can be defined by hand –No rule about which direction vertex normals must face (see next slide) Vertex normals used for lighting –To define the mesh curvature Widely useful elsewhere –Often just called normals

Multiple Vertex Normals Examples on last slide used one normal per vertex to define a smooth surface Can have multiple vertex normals on a single vertex To indicate that certain edges are “sharp” Will see this further when we cover lighting