Oblique projection

Figure 3.6 A sawhorse representation and a Newman projection of ethane. The sawhorse representation views the molecule from an oblique angle, while the Newman projection views the molecule end-on. Note that the molecular model of the Newman projection appears at first to have six atoms attached to a single carbon. Actually, the front carbon, with three attached green atoms, is directly in front of the rear carbon, with three attached red atoms.

Conformational isomers are represented in two ways, as shown in Figure 3.6. A sawhorse representation views the carbon-carbon bond from an oblique angle and indicates spatial orientation by showing all C-Tl bonds. A Newman projection views the carbon-carbon bond directly end-on and represents the two carbon atoms by a circle. Bonds attached to the front carbon are represented by lines to the center of the circle, and bonds attached to the rear carbon are represented by lines to the edge of the circle. 

Figure 16-16. Molecular packing of Oocl-OPV5 in the crystal lattice. Lett oblique view of (he unit cell of Oocl-OPV5 right projection of the unit cell on a plane perpendicular to the ci-axis.

The dimer chains of Ca -ATPase can also be observed by freeze-fracture electron microscopy [119,165,166,172-174], forming regular arrays of oblique parallel ridges on the concave P fracture faces of the membrane, with complementary grooves or furrows on the convex E fracture faces. Resolution of the surface projections of individual Ca -ATPase molecules within the crystalline arrays has also been achieved on freeze-dried rotary shadowed preparations of vanadate treated rabbit sarcoplasmic reticulum [163,166,173,175]. The unit cell dimensions derived from these preparations are a = 6.5 nm b = 10.7 nm and 7 = 85.5° [175], in reasonable agreement with earlier estimates on negatively stained preparations [88]. 

Fig. 7. Projection view of negatively stained Ca -ATPase crystals in sarcoplasmic reticulum solubilized with Ci2Eg (2 mg/mg protein) in the standard crystallization medium. The prominent large spacing is the half-period of the a cell dimension. Striations oblique to this direction are the (1,1) and the (-1,1) periodicities. Magnification, x 308 000. From Taylor et al. [156].

Fig. 2. Soft tissue imaging can be accomplished in pre- and postnatal specimens by micro-CT. Top rowdepicts a maximum intensity projection or WIIP view of an El 5 mouse embryo left) and PO mouse pup (right). Bottom rowdepicts a 2D slice view, technically termed a reformat, of the same El 5 (left) and PO (right) from the data set depicted above. The soft tissue features can be seen along with skeletal signal as volumetric or two-dimensional slices of the specimen in all anatomical viewing planes, including oblique angles, if desired.

Rotational Symmetry of 2D Lattices. Each of the five lattices has rotational symmetry about axes perpendicular to the plane of the lattice. For the oblique lattice and both the primitive and centered rectangular lattices these are twofold axes, but there are several types in each case. The standard symbol for a twofold rotation axis perpendicular to the plane of projection is . In the case of the square lattice there are fourfold as well as twofold axes. The symbol for a fourfold axis seen end-on is For the hexagonal lattice there are two-, three-, and sixfold axes the latter two are represented by a and , respectively. In Figure 11.4 are shown all of the rotation axes possessed by each lattice. 

Other common methods for representing the three-dimensional structures of molecules include Newman projections for showing conformational relationships and sawhorse figures. Newman projections look down a carbon-carbon bond so that the front carbon, designated by a circle, obscures the carbon directly behind it. Valences (bonds) to the front carbon extend to the center of the circle, while bonds to the rear carbon stop at the circle. Sawhorse projections have the carbon-carbon bond at oblique angles, which attempts to represent a perspective drawing of the molecule. Thus for 2-chloro butane, if one chooses to examine the 2,3 bond, then the sawhorse and Newman projections would be... 

Fig. 18.2. a,b Normal venous anomaly in a 3D phase contrast venous angiogram performed at 1.5 T. c,d 3D phase contrast venous angiogram in a patient with idiopathic intracranial hypertension displayed in different projections. The bilateral short stenoses (arrows) are well shown by MR venography, e digital subtraction angiogram in an oblique projection. Confirmation of the obstructed vessel lumen on both sides (arrows), but the finding at this location can only be demonstrated on special projections... 

Unless the sets V and 0 span the same space, the operator Q is usually not self-adjoint and it may hence be described as a skew projector associated with an oblique projection of nonorthogonal character. It is immediately seen that it has the particular properties... 

Figure 2. Unit cell of PdF, projected on the tt-b plane (viewed obliquely) Pd, small circles marked with the fractional z coordinate along c, F, large circles.

Figure. 7.10 The P-PCS membrane system in scaleworm photocytes (a)-(c) and lens (d). (a) A section identified as the [100] projection (mid to left regions of the section). It is best understood as cut between the 100 and the (llO) plane, at a distance of about 1/2, 0-1/2, 0, relative to the origin, (b) Complex projection along the [332] and the [553] directions. The latter is generated at a distance of approximately 5/4, 5/4,1/4. (c) Section cut normal to the [432] plane, (d) The P-PCS membrane in the lens of Lagisca extenuata. Oblique section corresponding to the [210], [221], and the [432] projections of the P-PCS. Figures (a)-(c) from [76], and figure (d) from [79], reproduced with permission.

Figure 7.13 The multicontinuous G-PCS membrane system identified as a part of SER retinal pigment epithelia cells of a river lamprey, (a) Projection along the [211] (left) and the [111] (upper right) directions. The lower ri t ows an oblique section which can be understood as cut between the (211) and the (111) planes. Scale bar 1 pm. (b) Eietail of the gyroid membrane showing the four approximately parallel membranes defining 5 different spaces. Scale bv 0.5 pm. Figs, (a) and (b) are reproduced from [92], with permission.

Figure 7.14 The SER associated gyroid membrane in adrenocortical cells of Salmo fitrio. (a) A rather thick section (> 0.5 times the unit cell edge) cut approximately normal to the 211) plane. Note the hatched low amplitude sinusoidal pattern, in between the large amplitude sinusoids. Compare this with Fig. 7.6(c) and its corresponding computer generated projection, (b) A typical oblique section of the gyroid membrane. From [57], reproduced with permission.

A sawhorse representation and a Newman projection of ethane. The sawhorse projection views the molecule from an oblique angle, while the Newman projection views the molecule end-on. 

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