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Table 7"SIGGRAPH '89, Boston, 31 July-4 August, 1989
trees at the branch tips. Obviously, with this method, the ramification matrix of the generated tree could be more different than the one given. An example is visualised in Figure 18 which seems to simulate an herbaceous plant.
6. The leaf model
In order to obtain realistic drawings (Figures 22 to 27), we describe herein a model for drawing leaves. Such an efficient and fast algorithm is primordial to draw trees with hundreds of branches and then thousands of leaves.
6.1 The leaf drawing model
As shown in Figure 19, a leaf is made of an arborescence of "veins" immersed in the limb (body of the leaf once we have removed the veins). In a first approximation, the leaf is supposed to be flat and the underlying arborescence is a ternary tree except at the leafstalk (or petiole) where the arity can take any odd value. Below, we call main vein a sequence of central edges issued from the petiole, and secondary vein, an edge issued from a main vein. We consider in the following that there is no deeper veins than secondary ones. A lobe is then constituted of a main vein, its secondary veins and the border surrounding the part of the limb associated to this main vein (border whose geometry is not specified at this level : polygonal, smoothed, far from or near the terminal nodes).
The topological leaf structure is defined by :
1) The number of lobes;
2) The "size" of the leaf, given by the number n of nodes on the central main vein. The number of nodes on each other main vein issued from the petiole being a simple function (usually decreasing) of n and of the "distance" between this vein and the central one.
The geometrical leaf structure is then obtained by defining:
1) The positions of nodes with the help of 3 parameters ; two angles A and 6 and a length law L for arborescence edges.
Ä is the angle between two successive main veins, and 6 is the angle between a secondary nervure and its main vein. The length L(d) of an edge is an increasing function of the "depth" d of the edge (d is the distance between the initial vertex of the edge and the final vertex of the main vein of the lobe). Three laws have been tested (the last one giving the best results) : (1) L(d) = Cte, (2) L(d) = A*d + B, (3) L(d) = A*Log(l + d)+ B. If two veins cross, one of the end segments is deleted.
2) The border of each lobe as a polygon joining the root and points in the continuation of the veins determined from the terminal nodes by a multiplicative factor.
The limb border is then the polygon obtained by joining parts of each lobe border belonging to this limb border (see Figure 19). This polygonal representation is sufficient when the leaves are drawn with a little size on the tree (seen from far, see Figures 22 to 25), allowing a fast drawing algorithm of a tree with its foliage.
In the foliage of a tree, leaves being mixed, it is necessary to distinguish them by various colors. A very realistic aspect is obtained by gradation of colors (see Figure 21). To this aim, each lobe color gradation is obtained by two operations : (1) an affinity whose axis is the lobe main vein, whose direction is the one of the secondary veins and whose ratio varying between 0 and 1 gives the color gradation by a function ; (2) an homothety whose center is the root of the arborescence.
When we need to represent a realistic leaf, the limb border must be smoothed. This is done by interpolation with the help of cubics of each lobe border polygonal segment (each segment is replaced by a cubic which is given by its two extremities and two associated tangent vectors, see Figures 19,21).
Such a method for modeling leaves using an underlying tree has already been introduced by Prusinkiewicz et Al. . They use L-systems. As in our method for generating trees, our leaf drawing algorithm separates growth and topology, the
topology being the dominant feature from which is issued the geometry. Our model differs from the one of P. Lienhardt  by using simple combinatorial tree structures instead of planar maps. This allows a faster drawing algorithm issued from an easy leaf topological modeling while obtaining a large diversity of forms (see Figure 21). The method proposed by Bloomenthal  allows the visualization and the positioning of a digitalized leaf (the maple leaf) with the help of an added polygonal structure for a 3D aspect simulation. Tbe method is quite different from ours : his aim is not to propose a general leaf model. Other fractal approaches appear, given in Oppenheimer , in which the external boundary shape of the leaf is the limit of the recursive fractal growth of the internal veins, and in Demko et al.  who generates leaves as fractal sets, using iterated function systems.
Central Main Main veins vein Limb
Figure 19. Leaf geometry.
6.2 Spot models
A faster algorithm can be obtained by replacing real leaves by spots. These spots can merely constitute a cloud of color points (two side background trees in Figure 27), or polygonal spots instead of points (the two foreground trees in Figure 27).