Merle Johnson of the City of Ann Arbor for permission to use City of Ann Arbor base maps and aerials in this article.

Karen Hart, Planning Director, and Chandra Hurd, Planning Department, City of Ann Arbor, for files concerning building height in the downtown.

Matthew Naud, Environmental Services Coordinator and Emergency Services Coordinator, City of Ann Arbor.

Prof. Peter Beier, Director 3D Laboratory, Media Union, The University of Michigan and his staff members Lars Schumann and Brett Lyons.

Thus, there is a need to have mechanisms to create continuing economic development within the city.  One way is to increase the stock of housing and space for commercial and other establishments in support of that housing.  This path is all the more attractive in light of enduring interests in reducing "sprawl" and in preserving open space in the more rural surrounding lands.  In a city with few remaining empty buildable lots, this approach seems to offer few alternatives, the most obvious of which is to increase the density of dwelling units within the city.  When density increases are proposed in established residential neighborhoods there is often loud and long public objection from residents of those neighborhoods.  There may also be serious environmental considerations, as well.  Few residents, however, seem to object to increasing density in the downtown:  many who already live in the downtown moved there with an acceptance of taller buildings. Residents of the city who do not live in the downtown often seem not to care about the idea of increasing density in the downtown.  What people do seem to care about, however, is what an increase in downtown residential density may mean to the character, appearance, and feeling of the downtown:  to its skyline and to the pedestrian experience.  To some, an 18 story building is a visual blight on the skyline that provokes negative comment every time it is viewed; yet, others note that they have become accustomed to it and view it as an old, familiar friend.  Building height can be a source of substantial dispute.

Inventory of the Vertical City
Prior to considering new tall buildings, it seems appropriate to create an inventory of existing buildings in the downtown area.  (In Ann Arbor, the "downtown" generally refers to the "Downtown Development Authority" or DDA:  a state-enabled authority that can capture increases in taxable value to pay for improvements within the defined boundaries.)  To create this inventory, building footprints were digitized from high quality aerial flown in 2002.  Heights were assigned to buildings based on information from the City of Ann Arbor Planning Department (only partially complete).  When the building footprints are sorted out according to height it becomes possible to visualize how the taller buildings are arranged with respect to the shorter buildings.  Figure 1 shows an animation of this pattern.  In that animation the reader has an opportunity to study different layers of downtown space in relation to a plain backdrop and finally to an aerial of the city.
 

Figure 1.  Animation of existing building height in downtown Ann Arbor, Michigan.

The evidence of Figure 1 suggests that buildings of 1, 2, and 3 stories are common in the downtown.  Indeed, casual conversations with individuals from around town suggest that no one objects to buildings of any of these heights.  One might wonder if that is because they somehow fit a sense of Ann Arbor well or if that is because they are prevalent and people become accustomed to them.  In any event, one might imagine an ordinance which allows three stories "by right" on any downtown parcel.  The question then becomes, how high elsewhere on prime parcels?  For this question one might look to the spacing pattern of existing buildings taller than three stories.  Tall buildings adjacent to other tall buildings can create wind tunnels and block wide channels of light.  Tall buildings built lot line to lot line may present those as well as other unwelcome effects.

The Floor/Area Ratio as an Urban Planning Tool

The problem of where to locate tall buildings, with sensitivity to existing building types on adjacent and nearby lots, is a difficult one.  In Ann Arbor, building height is currently limited by "floor area ratio" (FAR).  The FAR is calculated as the ratio of floor area in a building divided by parcel area, times 100.  If a given parcel has an FAR of 100 assigned to it, then a building footprint built lot line to lot line may have a height of 1 story.  If a parcel has an FAR of 200 assigned to it, then a building footprint built lot line to lot line may have a height of 2 stories.  Similarly, an FAR of 300, assigned to a parcel, yields a building of height 3 stories covering the entire parcel.   Thus, on a parcel with an FAR of 300, one might, instead, build a building on half of the lot area but of height six stories, or on a third of the lot area but of height 9 stories.  On the same parcel, a 30 story building could be built only if its footprint covered one tenth of the land area of the parcel.

The FAR provides a height limit based on the size of foundation needed to support a tall building.  It also offers subtle encouragement for preserving some amount of open space and visual variation in the region to which it applies.  The drawback is that a tall building may get built with no regard to the broader context of how that new building will fit in with existing buildings on the surrounding parcels.  A possible side effect of using FAR (alone) to limit height is that it might encourage parcel amalgamation by large developers, thereby driving out desired local small business owners.  [Note:  in Ann Arbor, there are also "premiums" designed to encourage residential construction, and other uses viewed as "desirable" in the downtown; these allow an increase in FAR.  They will not be covered in this discussion as they introduce no new theoretical issues--just complexity of detail.]

The Floor/Area Ratio, a Closer Look:  The Hyperbola as an Urban Planning Tool

In a  recent article Claudia Iturriaga and Anna Lubiw consider the problem of labeling maps.  Because the current mapping environment is one that allows dynamic positioning of maps (zooming-in and panning), they consider the problem of non overlapping placement of text boxes to be one that is sufficient to solve with text boxes only at the perimeter of the map (with map content in the interior).  They note that if the aspect ratio of the label (ratio of height to width) is permitted to vary, with label area held constant, then labels can be fit together in a variety of patterns that will permit a balanced display of map and text boxes.  The requirement of constant label area ensures that a certain amount of text content is communicated; shape is permitted to vary.  Thus, if the label is viewed as having a fixed lower left corner, then the upper right corner varies along the track of the first quadrant of a rectangular hyperbola with origin at the lower left corner.  That is, if width is measured along the x-axis and height is measured along the y-axis, and the area of a label is fixed at K, then the equation describing the label is xy = K.  This latter equation is precisely the equation of a rectangular hyperbola in the first and third quadrants intersecting the line y = x at (K, K).

It is not a long conceptual leap to imagine the rectangular areas arranged around the perimeter of a rectangular map as being similar to the rectangular areas of building footprints arranged around a rectangular block of a downtown based on a gridded street system.  The idea of a rectangle with an elastic aspect ratio tracing out the path of an hyperbola is similar to the idea of Floor Area Ratio (FAR) discussed above.  From an abstract viewpoint, the FAR/100, or number of stories, times the parcel area serves as an envelope within which buildings may be built.  For example, if a parcel has area 100,000 square feet and an FAR of 300, then 300,000 square feet of floor area may be built on the parcel:  as a 3 story building lot line to lot line front, back, and sideways (green building in Figure 2); or, as a 6 story building with each floor having 50,000 square feet on half the parcel (yellow building in Figure 2); or as a 12 story building with each floor having 25,000 square feet on 25% of the parcel area (magenta building in Figure 2).  What is constant is the value, K = (FAR/100)*(parcel area).  If one graphs this function, with parcel area on the horizontal axis and FAR/100 on the vertical axis, the result is a rectangular hyperbola, xy = 300,000 (Figure 2).  Different masses of building in relation to land area result depending on the height one chooses.
 

Figure 2.  Rectangle with elastic aspect ratio and lower left corner fixed at the origin traces out part of one limb of a rectangular hyperbola xy = 300,000.

When one abstracts away from the grid suggested by Figure 2, and focuses instead on the hyperbola, it is possible to extend the analysis to the more global scene of the entire DDA and to the issue of building mass in relation to land area.  Thus, consider that the x-axis units are now percent area in the downtown; then, the right-hand limit of the hyperbola is 100% of the land in the DDA.  Under these assumptions, what the hyperbola of Figure 2 now says is that 100% of the DDA may be covered with 3 story buildings:  that a 3 story building may be constructed, by right, anywhere within the DDA.  It also says that 50% of the land area in the DDA may be covered with 6 story buildings, or that one quarter of the land area in the DDA may be covered with 12 story buildings, or that 10 percent of the land in the DDA may be covered with 30 story buildings.  The use of the FAR to govern building height may play our at a regional (DDA) level as well as at a local level of the individual parcel.  The hyperbola captures the FAR in a systematic manner and it does so at all scales, from local, to regional, to global.  It does not reflect planning and geographic elements that the FAR does not capture such as (but not limited to)  heights of neighboring buildings and other adjacency considerations, historic preservation issues, shadow or wind tunnel effects and other quality of life issues, or lateral or upper story setback concerns.  Issues such as these require the human elements of judgment and common sense.  The mathematical implementation can do much, but not all; it is a tool of humans, not a replacement for human thought (although numerous abstract connections remain to be probed:  from cartography, to urban planning, to the Zipf rank-size rule and the lectures given by Michael Batty at The University of Michigan and Eastern Michigan University in the spring of 2003).

The principles set forth here, would enable one to consider the total mass of building square footage permitted according to FAR, independent of municipality and local concerns.  Subtracting the actual built up area from that would give an estimate of the remaining mass that could be built, by right, according to code.  Within that remainder, one might calculate how many more 3 story buildings could be built; how many more 6 story buildings; how many 12 story buildings (or whatever height in whatever units).  Such a strategy can completely characterize the mass of building in relation to land area and may suggest a basis for the control of that mass, especially when one decides what future is desired and works back from that to create ordinances and code that will lead to that desired outcome (an approach similar to that take by others, as for example by people at ChicagoMetropolis2020).  It offers, however, no guidance as to where tall buildings might be placed in relation to each other or in relation to existing structures, as to which parcels might contain tall buildings, as to wind, light, and sound issues, and as to a host of other qualitative issues.  Other approaches might involve a guide to the spacing of buildings (forthcoming), buffers around existing buildings as zones of limited height, or legislated design standards.  It is for creative needs such as these, to be superimposed on measures of sheer mass or quantity that can be captured generally as mathematical and geographical propositions, that cities require the service of professional planners and a host of municipal authorities and support personnel.

Beyond the Floor/Area Ratio:  Virtual Reality as an Urban Planning Tool.

     Virtual reality, the envisioning of alternative three-dimensional scenarios on a computer screen, offers to decision makers the capability to see how the massing of buildings and the general design of the urban landscape might look with various changes.  In the case of Ann Arbor, that might mean envisioning the downtown with new tall buildings in a three-dimensional model that can be viewed at the pedestrian level:  as a virtual landscape that can be navigated on the computer screen by City Council members as they sit with laptops in Council Chambers or by members of the public as they sit at home or in public libraries using computers with internet connections.  Part I of this topic showed virtual reality of the downtown based on

• VR 1:  parcels were extruded to form chunky buildings that filled entire parcels, lot lines to lot lines, with height assigned by FAR and zoning ordinance (C1A, 200% FAR; C1A/R, 300% FAR; C2A, 400% FAR; C2A/R, 300% FAR; C2B/R, 300% FAR).
• VR 2:  parcels were extruded to form chunky buildings that filled entire parcels, lot lines to lot lines, with height assigned by records from the Planning Department of the City of Ann Arbor.

The following sequence of interactive maps, made using the ImageMapper 3.3 extension to ArcView, shows the results, using maps and aerials in various combinations:

• I-Map 1:  Click here for a link to an interactive map showing building footprints and height (on mouse-over) as well as building address and street names (on mouse-over).  Parcel boundaries are shown on the underlying aerial and on the green Downtown Development Authority (DDA) area.  The Allen Creek floodway (underground) and flood plain are shown, shaded, respectively in blue and turquoise.  Click on a building or a street to see associated entries in the underlying database.
• I-Map 2:  Click here for a link to an interactive aerial showing parcel boundaries, zoning, building height (on mouse-over), and street name.  DDA outline, only, is shown in light yellow so the user may zoom in to get a closer view of the aerial within the DDA (up to 800% enlargement--can see cars clearly).  The Allen Creek floodway (underground) and flood plain are shown, outlined, respectively in blue and light blue; again, because the shading is removed, the viewer may look at the content of the floodway/floodplain in greater detail than above. Click on a building or a street to see associated entries in the underlying database.
• I-Map 3:  Click here for a link to an interactive aerial showing zoning boundaries in the downtown, zoning type (on mouse-over), building height (in the "zoneht" record of the database), and street name.  Click on a building or a street to see associated entries in the underlying database.

On November 9, 2003, City Council Member Jean Carlberg (and Mayor ProTempore, Planning Commissioner, and member of the Ordinance Revisions Committee), City Council Member Joan Lowenstein, City of Ann Arbor Planning Director Karen Hart, and former City Attorney (on two occasions) Jerold Lax, visited the GeoWall (with the author and others, a total of 14) at The University of Michigan's 3D Laboratory at the Media Union (Dr. Peter Beier, Director).  At that time, they had the opportunity to view the files above at a scale that permitted them to feel as if they were walking among the buildings.  Each was given the map displayed in Figure 3 and an earlier version of the commentary following the map.  The red building on the map in Figure 3, at the southeast corner of Fifth and Huron Streets, is a location mentioned as a possible site for a new tall building by Ann Arbor Mayor John Hieftje (in personal communication with the author and elsewhere).  The commentary following the map enumerates the steps taken to build a virtual structural base of the downtown to use as a model to consider density/height issues in the downtown.
 
 

Figure 3.  Map handed out to participants in the GeoWall display of November 9, 2003 at the Media Union of The University of Michigan.

 

Comments from the meeting from November 9, 2003 and subsequent follow-up:

• Council Member Carlberg noted that she might also wish to know more about where the shadows of new buildings might fall.  Lighting changes are difficult to model in VR; however, with aerials that show existing building shadows, it is not hard to imagine where shadows of new buildings might fall.  Thus, in the

one sees a red square on a parking lot corresponding to the location mentioned as a possible location for a tall building by Mayor Hieftje.  The buildings around it cast shadows that extend almost across the street.  A new building on the red square, of height greater than adjacent buildings would cast a shadow on both sides of the street.  Shadow position is important when considering budgetary allocations from the city's street tree escrow.  It is also important in creating a positive pedestrian experience in the downtown.

• Council Member Lowenstein commented to the author that the files above were, with navigation aids added, probably enough to be quite useful to City Council.  Both she and Planning Director Hart noted their utility in considering issues involving height in the downtown as they relate to a recent city initiative to increase the residential population in the downtown.  She also noted that the addition of callouts (notes) that show which buildings might contain hazardous materials, or similar information, might be helpful to firefighters and other emergency first responders.  Two-dimensional interactive maps or aerials may well be sufficient for a hazardous materials inventory.

• An I-Map based on an aerial might offer one approach.  On the

the mouse-over callouts shows the building address for three locations.  Click on a location to reveal elements of the database associated with each site.  In seeing all buildings simultaneously one gets an immediate picture of adjacency patterns: for example, a fire in one building may need immediate containment on the eastern edge to prevent spread to an adjacent building on the east containing volatile material. Careful database construction is critical:  the mapping, in this case, is easy in relation to the database construction.

• A very simple approach might simply employ Adobe Photoshop (version 7.0 was used here) to work with a high quality aerial photograph of the City.

In the

note files and voice files have been added to City Hall, to 219 S. Main, and to the central quadrangle (the "Diag") of The University of Michigan.  Thus, emergency workers might have not only the benefit of reading notes attached to buildings that specify the locations of hazardous materials, but also the capability to hear voice transmissions of such locations when already in a tight spot.  The drawback to this style of approach is that it requires the user to download the file and open it in Adobe Photoshop (or use some similar strategy to read the notes).  If, however, the emergency management team already has Photoshop loaded on laptops, this is not much of a disadvantage.  Indeed, it might be viewed as an advantage in file security given that it does not play directly on the Internet.
• Planning Director Hart, noted in addition, the importance of modeling upper story setbacks as a next step.  She also suggested possible specific locations in the downtown where VR might be particularly helpful, including in the modeling of various aspects of long-standing plans for a renovation of governmental space.  As convincing and as helpful as virtual reality can be, it is however, only virtual.  When one walks away, it remains only in the mind.  Another exciting technological tool that the group saw is the 3D "printer" that creates true 3D objects representing the experienced virtual reality.  Hart also noted that she could see numerous uses for this tool.  Indeed, sometimes the end desired suggests the process to get there, not only in master planning and other forms of planning, but also in the tools used in planning.

The display below presents the final experiments in this set (given to Ann Arbor City Council in December of 2003) as the first in a series of possible 3D mapping  tools to aid in making a variety of difficult decisions:  for Ann Arbor as well as more globally.  It includes parcels extruded from building footprints, with the sun set in the south at a "low" setting, using an invisible topographic base created from a TIN made from a topographic map with a contour interval of 5 feet.  Buildings have been adjusted using a z-factor of 0.3048.  It also includes street labels that appear as one moves around at a local level as well as navigation aids (click in the lower left corner of Cosmo Player) of assigned camera viewpoints.  These, coupled with using the "driving" capability of Cosmo Player, help in getting around the virtual downtown so that one does not get lost in the space of virtual Ann Arbor!
 
 

Next steps include:

• Field checking of building heights
• Modeling of upper story set backs

• Thinning of file size based on scale.
• Produce a number of other files based on various lighting possibilities.
• Introduce cars along the streets, pedestrians on the sidewalks, and so forth.
• Model the weather (colleague John D. Nystuen suggested modeling a snow storm).  Nystuen also suggested modeling the underground infrastructure.
• Consider how practical, day to day elements of decision making might be aided.
• Might VR files serve to replace the model consideration in the PUD zoning?
• If so, what sort of ordinance revision would be necessary and what legal ramifications might there be in such a consideration or in related ones?

More generally, what are the legal questions involved in using VR as a planning and emergency management tool; do they differ from those associated with using 2D analysis for such purposes?

• her colleagues on the City of Ann Arbor Planning Commission (Sandra Arlinghaus (Chair), Kevin McDonald (Vice-Chair), Scott Wade (Secretary), Braxton Blake, Jean Carlberg, Kristen Gibbs, Christopher Graham, William Hanson, and Steve Thorp);
• the Ordinance Revisions Committee of that Commission (Hanson, Chair; Carlberg, Arlinghaus, Blake);
• the City of Ann Arbor Planning Department staff (Karen Hart, Planning Director; Wendy Rampson, Coy Vaughn, Donna Johnson, Jeff Kahan, Chandra Hurd, Alexis Marcarello, Christopher Cheng, and Matthew Kowalski);
• Merle Johnson, City of Ann Arbor, Information Technology Services;
• Heather Edwards, Historic District Preservation Coordinator, City of Ann Arbor;
• Matthew Naud, Environmental Coordination Services Director and Emergency Management Director, City of Ann Arbor
• John D. Nystuen, Professor Emeritus, Taubman College of Architecture and Urban Planning, The University of Michigan
• Peter Beier, Professor of Engineering and Director, 3D Laboratory, Media Union, The University of Michigan.
• the Mayor of Ann Arbor, His Honor, John Hieftje

REFERENCES

• Ann Arbor Zoning Ordinance, Chapter 55, Ann Arbor City Code, pp. 36-38.
• Arlinghaus, S.  Ann Arbor, Michigan:  Virtual Height Experiments, Solstice:  An Electronic Journal of Geography and Mathematics, Volume XIV, No. 1, 2003, Institute of Mathematical Geography.
• Batty, M.  Lecture series on Zipf Rank-Size Rule, The University of Michigan and Eastern Michigan University, Spring, 2003.
• Chicago Metropolis 2020
• Churchill, R. V.  Complex Variables and Applications, 2nd Edition (1960), New York:  McGraw-Hill.
• Iturriaga, C. and Lubiw, A.  Elastic labels around the perimeter of a map.  Journal of Algorithms, 47 (2003) 14-39.

• ArcView GIS, v. 3.2, with Spatial Analyst Extension and 3D Analyst Extension.  All from ESRI (Environmental Systems Research Institute, Redlands, CA). http://www.esri.com/
• ImageMapper 3.3, Alta 4, http://www.alta4.com/
• Microsoft Windows XP, http://www.microsoft.com/
• Cosmo Player, http://ca.com/cosmo/
• Adobe PhotoShop and Adobe ImageReady, versions 7.0.